Tensile fatigue behavior of fiber-reinforced cementitious material with

Construction and Building Materials 178 (2018) 349–359

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Tensile fatigue behavior of fiber-reinforced cementitious material with high ductility: Experimental study and novel P-S-N model Bo-Tao Huang, Qing-Hua Li ⇑, Shi-Lang Xu, Bao-Min Zhou Institute of Advanced Engineering Structures and Materials, Zhejiang University, Hangzhou 310058, China

h i g h l i g h t s The stress-control tensile fatigue behavior of UHTCC is investigated. Four stages were observed in the evolution curve of fatigue deformation. Smooth and rough areas can be distinguished on the fatigue failure surfaces. Novel P-S-N models based on a modified S-N relation are proposed. Using high-strength high-modulus fibers to replace partial PVA fibers in UHTCCs may improve the fatigue performance.

a r t i c l e

i n f o

Article history: Received 21 February 2018 Received in revised form 15 May 2018 Accepted 22 May 2018

Keywords: Tensile fatigue Fiber-reinforced ECC SHCC Stress level P-S-N model

a b s t r a c t Fiber-reinforced cementitious material with high ductility is a cement-based material with strainhardening behavior under tension, and has potential application in structures sustaining fatigue loads. In this study, the tensile fatigue behavior of this material at various stress levels (S = 0.90, 0.85, 0.80, 0.75, 0.70, and 0.65) is investigated with the stress ratio of 0.1. The fatigue crack pattern, deformation, failure surfaces, and fiber failure modes are analyzed. Four stages are observed in the evolution curve of fatigue deformation. This is different from the three-stage curve of conventional concrete. ‘‘Smooth” and ‘‘rough” areas are distinguished on the fatigue failure surfaces with different fiber failure modes. Emphasis is placed on the development of a novel probabilistic model. On the basis of the initial distribution of static strength, P-S-N (probability of failure-stress level-fatigue life) models are proposed for a reliable application of this material. Moreover, a suggestion to improve the fatigue life of this material at low stress levels is provided. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction Fatigue of concrete is a process of mechanical deterioration caused by repeatedly applied loads. The fatigue of concrete materials first attracted interest with the construction of reinforced concrete bridges over 100 years ago. The fatigue behavior of concrete structures in modern infrastructures has been receiving increasing attention from researchers. On the one hand, with an increase in demand for high-speed transportation systems (e.g., highways and high-speed railways), especially in developing countries like China, the concrete structures in such infrastructures are required to sustain a larger amount of repeated loading in the same service period. On the other hand, transportation infrastructures in urban areas have to sustain a larger amount of traffic loading with an ⇑ Corresponding author. E-mail addresses: [email protected] (B.-T. Huang), [email protected] (Q.-H. Li), [email protected] (S.-L. Xu), [email protected] (B.-M. Zhou). https://doi.org/10.1016/j.conbuildmat.2018.05.166 0950-0618/Ó 2018 Elsevier Ltd. All rights reserved.

increase in the urban population, since the proportion of the urban population is expected to increase to 66 percent by 2050 according to the World Urbanization Prospects by the United Nations [1]. Thus, knowledge of the fatigue behavior of concrete materials is truly important in modern construction. In recent decades, fiber-reinforced concrete has been investigated to improve the performance of concrete. The inclusion of fibers can have a beneficial effect on the fatigue performance of concrete, and this effect is more significant in fiber-reinforced cementitious material with high ductility. This material, which was proposed by Li and Leung [2], is reinforced by randomly distributed short fibers, and exhibits strain-hardening behavior under tensile loading. These materials are known as Engineering Cementitious Composites (ECCs) [3], Strain-Hardening Cementitious Composites (SHCCs) [4], and Ultra-High Toughness Cementitious Composites (UHTCCs) [5]. UHTCC shows a longer fatigue life and larger deformation than ordinary concrete [6–11], and it has potential applications in high-durability structures [12–19].

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B.-T. Huang et al. / Construction and Building Materials 178 (2018) 349–359

Several studies have been carried out to investigate the fatigue behavior of UHTCCs under tensile loadings [10,20–22]. Matsumoto et al. [20] first conducted fatigue tension experiments on UHTCCs under a constant displacement amplitude, and the bridging stress degradation under fatigue loading was investigated. Determinations of bridging stress degradation relations by micromechanicsbased and experimental-based approaches were proposed. Matsumoto et al. [21] investigated the displacement-control tensioncompression fatigue behavior of UHTCCs, and proposed fitting curves based on a Weibull distribution to describe the bridging stress degradation. Müller and Mechtcherine [10] performed tension-swelling and tension-compression fatigue tests, with the upper reversal point controlled by a given deformation increment or load level and the lower reversal point controlled by a given load value. For each fatigue loading scenario, the number of load cycles to failure were investigated, and four failure modes were identified. Huang et al. [22] conducted fatigue tension tests on precracked specimens with a constant crack opening displacement amplitude, and the relationship between the crack bridging stress and the number of loading cycles was established. UHTCCs can be applied in the tensile zone of reinforced concrete structures depending on the tensile strain-hardening response. The tensile strength of UHTCCs in a cracked state is considered for components design under monotonic loadings [14–16], while that of conventional concrete is always neglected. For particular fatigue loadings, UHTCCs in a cracked state are still able to sustain dozens to millions of loading cycles [10]. This indicates that the tensile strength of UHTCC can also be considered for concrete structures under fatigue loadings. In the fatigue design of structures, the stress level S (i.e., the ratio of the maximum fatigue stress rmax to the average static strength r0-ave) and the stress ratio (i.e., the ratio between the minimum fatigue load to the maximum fatigue load) are the determining factors. Thus, a solid knowledge of the tensile fatigue behaviors of UHTCCs at various stress levels, especially fatigue life, is obviously important for its practical application in modern infrastructures such as transportation systems. However, to date, few studies have investigated the influence of the stress level on the tensile fatigue behavior of UHTCCs. A systematic investigation of this problem, especially with regard to UHTCCs at low tensile stress levels, is essential for the design of structures using UHTCCs. In this study, the tensile fatigue behavior of UHTCCs at various stress levels (S = 0.90, 0.85, 0.80, 0.75, 0.70, and 0.65) is investigated. The fatigue crack pattern, deformation, failure surfaces, and fiber failure modes are analyzed. On the basis of the initial distribution of static strength, P-S-N (probability of failure-stress level-fatigue life) models are proposed for a reliable application of this material. Moreover, a suggestion to improve the fatigue life of UHTCCs at low stress levels is provided. Finally, relevant conclusions are drawn.

2. Experimental program In this experiment, a UHTCC was produced using cementitious binders, fine silica sand, water, polycarboxylate superplasticizer, and polyvinyl alcohol (PVA) fiber. The proportions of the matrix were cementitious binders:water:fine sand = 1:0.24:0.6 [8,9]. The ratio of the superplasticizer to the cementitious binders is 0.14% (by weight). The maximum aggregate size of the silica sand is 300 lm. The PVA fiber was 40 lm in diameter and 12 mm in length, and the fibers in the matrix were 2% of the UHTCC volume. The mixing process of UHTCC was as follows: 1) mixing the cementitious binders and fine sand for 2 min, 2) adding water and mixing for 2 min, 3) adding the superplasticizer and mixing for 5 min, and 4) adding the PVA fibers and mixing for 5 min. Cuboid specimens

(60 mm in length 60 mm in width 450 mm in height) with the same matrix were prepared. The specimens were demolded 72 h after they were cast and were cured up to 28 days (temperature: 20 °C, relative humidity: 95%). Then, they were laid in an ambient environment for three months before testing. The setup for the tensile tests shown in Fig. 1 (a) was used in the monotonic and fatigue tests. Generally, the tensile deformation of UHTCC can be measured by linear variable differential transformers (LVDTs) fixed to specimens. In such case, the initial part of stress-strain curve shows a stiff linear branch before cracking. In the fatigue tests of the UHTCC, the locations of major cracks were always random, and the major cracks were located inside or outside of the gauge length of the LVDTs, even exactly at the contact point of the LVDTs. However, a certain proportion of the fatigue life of the UHTCC occurred after the major crack appeared. This means that the tensile fatigue deformation measured by the LVDTs fixed to the specimens might be affected by the location of the major crack. Moreover, the UHTCC showed multi-cracking behavior during the tensile test. Extensive cracking may also lead to disturbances at the contact points of the LVDTs [23]. Thus, in this experiment, the displacement between the two clamping ends were measured at a 160-mm gauge length, instead of LVDTs fixed to the specimens. In this case, the end effect may have an influence on the measured tensile displacement. That is, some micro cracks may form around the clamping ends of the specimen. Hence, the linear branch could not be observed in the initial part of the stress-displacement curve of this study. It needs to be pointed out that during the monotonic and fatigue tests, the major cracks of most specimens were located in the gauge length. Those that were not, were eliminated from the test results. The monotonic tests were performed using displacement control at a constant rate of 0.20 mm/min, and the tensile strength of the UHTCC is listed in Table 1. The average tensile strength r0ave is 3.0 MPa and the standard deviation is 0.3 MPa. It should be pointed out that the tensile strength of the UHTCC is calculated by the maximum load during the monotonic test. For fatigue tests, a constant-amplitude fatigue load was used. The loading of the tensile fatigue test is presented in Fig. 1 (b). Six stress levels (S = 0.90, 0.85, 0.80, 0.75, 0.70, and 0.65) were considered in this investigation, the stress ratio was kept at 0.1, and the corresponding fatigue lives are also presented in Table 1. After the fatigue test, the selected pieces for the static and fatigue failure surfaces of the specimens were analyzed using a scanning electron microscope (SEM).

3. Tensile fatigue behavior 3.1. Crack pattern The typical crack patterns of specimens at various stress levels are illustrated in Fig. 2. One specimen was selected for static specimens (i.e., S = 1) and fatigue specimens (i.e., S = 0.90, 0.85, 0.80, 0.75, and 0.70). The visible cracks in the middle part of the specimens (i.e., 160 mm 60 mm) are marked using black lines. In the specimen under a static load, significant multi-cracking can be observed, and the cracks are uniformly distributed. In comparison, although the UHTCC specimens under fatigue loads also show multi-cracking behavior, there are fewer cracks in the specimens under fatigue loads than for the static one. This phenomenon may be related to the fatigue-induced changes of microscopic properties that lead to a reduction in the peak bridging strength and the complementary energy of fiber bridging [24]. It can be observed in Fig. 2 that the number of cracks in specimens under fatigue load decreases as the number of cycles increases (the stress level S decreases). At S = 0.90, a large number

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Fig. 1. Tensile tests: (a) test setup, (b) fatigue loading.

Table 1 Results of monotonic and fatigue tests. No.

1 2 3 4 5 6 7 8

Tensile strength (MPa)

2.619 2.792 2.794 2.944 3.000 3.061 3.428 3.481

Fatigue lives N at various stress levels S 0.90

0.85

0.80

0.75

0.70

0.65

646 1587 2133 2786 2790 3553 / /

1850 4175 5112 8862 10,005 14,944 / /

5859 6377 19,055 30,950 31,023 32,594 / /

18,125 20,096 29,707 54,904 64,180 133,807 / /

220,242 279,260 697,789 1,002,784 1,139,161 2000000* / /

2000000* 2000000* / / / / / /

Note: * means that fatigue failure did not occur when number of load cycles reached 2 million.

Fig. 2. Typical crack patterns of specimens at various stress levels.

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of cracks can be seen. These contribute to a larger tensile deformation. However, at S = 0.70, significantly fewer cracks can be observed, leading to a smaller tensile deformation. A similar phenomenon was observed in the flexural and tension-compression fatigue behavior [10,24,25]. In practice, the concrete structures of transportation infrastructures are always required to sustain multimillions of repeated loads. This means the structures are under fatigue loads with relatively low stress levels. The strainhardening characteristic of UHTCC becomes much less pronounced in this case, and this phenomenon is noticeable in the future application of this material. 3.2. Fatigue deformation The typical evolution curve of the tensile fatigue deformation of plain and fiber-reinforced concrete is an S-shaped three-stage curve [26–28]. The three stages are the rapid developing stage, stable developing stage, and failure stage. For a UHTCC, a typical evolution curve of tensile displacement of a fatigue specimen (S = 0.85, N = 4175) with respect to the ratio of load cycles is shown in Fig. 3. It should be noted that dmax is the displacement corresponding to the maximum fatigue load, and dmin is the displacement corresponding to the minimum fatigue load. In the left window of Fig. 3, a comparison of the average static displacement and fatigue displacement is presented. In general, the envelope of the tensile fatigue deformation of the UHTCC in Fig. 3 is also S-shaped. Four stages can be observed in the evolution curve, which is different from the three-stage curve of conventional concrete. Stage (I) is the rapid developing stage, similar to that of conventional concrete. Stages (II) and (III) are stable developing stages with different increasing rates. It can be observed that the increasing rate of Stage (III) is faster than that of Stage (II). In addition, the displacement range of one cycle in Stage (III) is much larger than that of Stage (II). Actually, the displacement of the turning point between Stages (II) and (III) is close to the displacement of the starting point of the strain-hardening branch under a static load. This can be seen in Fig. 3. This indicates that the differences between Stages (II) and (III) may be related to the strain-hardening response of the UHTCC. Stage (IV) is the failure stage, which may be related to the propagation of the major crack. For comparison, the stress-displacement curves of four cycles (i.e., the ratio of load cycles n/N = 0.02, 0.10, 0.50, and 0.90) are plotted in the left window of Fig. 3. These four cycles are located in the four stages of the evolution curve, respectively. It can be seen that the area surrounded by the stress-displacement curve

becomes larger when the number of cycles increases. This indicates that the energy dissipation of each cycle increases, and more cracks form in the UHTCC, as the specimen is closer to fatigue failure. UHTCC shows strain-hardening behavior after first cracking under tensile loading. In the stress-control tensile fatigue test, if the maximum stress is higher than the first cracking strength of UHTCC, the Stages (I) and (II) in Fig. 3 might disappear and only stages (III) and (IV) were left. In this case, the specimen would suffer considerable damage in the first load cycle, and the Stages (I) and (II) would end in this cycle. Hence, the fatigue life of UHTCC would also be much lower. If the maximum fatigue stress is lower than the first cracking strength, the four-stage fatigue deformation behavior could be observed. Besides, as the stress level decreases, the proportion of the Stages (I) and (II) would be higher and a longer fatigue life can be obtained. 3.3. Failure surface Typical failure surfaces under static and fatigue tensile loads are shown in Fig. 4. One static failure surface (i.e., S = 1) and five fatigue failure surfaces (i.e., S = 0.90, 0.85, 0.80, 0.75, and 0.70) are selected. These correspond to the specimens shown in Fig. 2. On the static failure surface, pulled-out PVA fibers are visible by the naked eye almost everywhere in the failure plane. However, on the fatigue failure surfaces, some ‘‘smooth” areas can be observed by the naked eye. These areas have very few PVA fibers, which are surrounded by dashed lines in Fig. 4. In addition, ‘‘rough” areas with large numbers of pulled-out PVA fibers, whose characteristic is similar to that of the static failure surface, can be found on the fatigue failure surface. Similar ‘‘smooth” areas are observed on the fatigue failure surface of metals such as steel [29]. Moreover, it can be seen that the ‘‘smooth” area becomes larger as the stress level decreases. At S = 0.90, the ‘‘smooth” area covers less than 1/10 of the failure surface. At S = 0.70, the ‘‘smooth” area covers about half the failure surface. In fact, the ‘‘smooth” areas form during the propagation of the fatigue-induced cracks. The UHTCC matrix at the edge of specimens with fewer constraints might crack more easily, and thus the fatigue-induced cracks possibly start from the edge of specimens. This may be the reason why most ‘‘smooth” areas in Fig. 4 are at the edges of failure planes. The ‘‘rough” areas may be considered as areas sustained during the last several load cycles. When the number of load cycles increases, fatigue-induced cracks propagate, and the average tensile stress in the ‘‘uncracked” areas increases. Once the average stress is equal to the tensile strength

Fig. 3. Evolution curve of fatigue deformation of UHTCC (S = 0.85, N = 4175).


353

Fig. 4. Failure plane of specimens under static and fatigue loads.

of the ‘‘uncracked” areas (the last fatigue cycle), then fatigue failure occurs. Thus, the failure during the last fatigue cycle is close to the static failure, and is the formation mechanism of the ‘‘rough” areas on the fatigue failure surface. In addition, as the stress level decreases, the tensile force on the ‘‘uncracked” areas decreases, and hence the ‘‘rough” areas become smaller. Fatigue-induced lamellar structures can be observed in the fatigue failure plane. This is shown in Fig. 5. The lamellar structures are marked with dashed lines. This phenomenon may be related to the multicracking behavior of the UHTCC and the propagation of the fatigue-induced cracks. For UHTCC, multiple cracks can be observed under tensile fatigue loading. As mentioned above, the propagation of fatigue-induced cracks would form the ‘‘smooth” areas. If the multi-cracking phenomenon occurs around the major crack and these fine cracks keep on propagating during the fatigue failure process, the fatigue-induced lamellar structure with

‘‘smooth” surface may be observed on the failure plane. To fully understand this phenomenon, further investigation is also required in the following work. 3.4. Microscopic investigation Selected pieces of the static and fatigue failure surfaces of the specimens were analyzed using SEM tests to obtain a better understanding of the formation process of the failure surfaces. SEM images of PVA fibers in the static and fatigue failure planes are shown in Figs. 6 and 7. In Fig. 6, two fiber failure modes, including a pulled-out one (Image 2) and ruptured one (Image 3), can be observed on the static failure surface. These two failure modes were reported in some previous investigations [8,30]. The SEM images of the ‘‘smooth” area on the fatigue failure surface are shown in Fig. 7. It should be remembered that very few PVA fibers

Fig. 5. Fatigue-induced lamellar structures on failure plane (S = 0.85, N = 4175).

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Fig. 6. SEM images of PVA fibers on static failure surface (‘‘rough” area).

Fig. 7. SEM images of PVA fibers on fatigue failure surface (‘‘smooth” area).

can be observed by the naked eye on the ‘‘smooth” areas. However, the PVA fibers can be seen in the SEM images (seen in Fig. 7). It can be observed that most PVA fibers are ruptured with very short ends out of the matrix. This phenomenon may be the reason why few PVA fibers can be observed with the naked eye in these areas, and it could be consider as a specific feature of the fatigueinduced rupture of PVA fiber. In addition, the details of the fatigue-induced ruptured fibers can be seen in the enlarged images (Images 2 and 3 in Fig. 7), and the striation-like patterns and lamellar structures can be observed on the failure end of the PVA fiber. It indicates that the fibers suffered from severe fatigue damage, and it could also be consider as a specific feature of the fatigue-induced rupture. Additionally, the fatigue-induced rupture of PVA fibers indicates that using fibers with a higher strength and modulus may improve the tensile fatigue behavior of a UHTCC. 4. Fatigue life and P-S-N models It has been reported that the fatigue life of concrete materials follows the Weibull distribution [8,9,31–33]. Recently, a probabilistic fatigue model based on the initial Weibull distribution of compressive strength was proposed to consider the frequency effect in plain and fiber reinforced concrete [34,35]. The fatigue lives of UHTCCs in Table 1 indicate a considerable scatter. Thus, it is desirable to introduce probabilistic concepts to analyze the reliability of UHTCCs under tensile fatigue loading. In this study, the effect of the stress level on the tensile fatigue life of a UHTCC is analyzed using the Weibull distribution. First, a two-parameter Weibull distribution is applied to describe the distribution of tensile strength and fatigue life of the UHTCC. Second, a modified S-N

relation is introduced to describe the relation between the distributions of tensile strength and fatigue life. Finally, P-S-N models of UHTCCs under tensile fatigue loads are proposed. 4.1. Distribution of tensile strength and fatigue life For the tensile strength of a UHTCC, the cumulative distribution function of a two-parameter Weibull distribution can be written as follows:

k0 ! r0 PF ðr0 Þ ¼ 1 exp k0

ð1Þ

where PF is the failure probability, r0 is the tensile strength of the UHTCC, k0 is the scale parameter, and k0 is the shape parameter. By fitting the tensile strength in Table 1, the values of k0 and k0 can be obtained: 3.1635 MPa and 9.2335, respectively. The corresponding coefficient of correlation r is 0.98, which indicates the tested tensile strengths of the UHTCCs follow the two-parameter Weibull distribution. For the tensile fatigue life of the UHTCC, the cumulative distribution function of a two-parameter Weibull distribution can be written as follows:

PF Nf

kN ! Nf ¼ 1 exp kN

ð2Þ

where N is the tensile fatigue life of the UHTCC, kN is the scale parameter, and kN is the shape parameter. Using the test results in Table 1, the scale and shape parameters of the fatigue lives of UHTCCs at various tensile fatigue stress levels were fitted using a

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B.-T. Huang et al. / Construction and Building Materials 178 (2018) 349–359 Table 2 Fitting parameters of Weibull distribution. S

kN

kN

r

NPF=0.5

0.90 0.85 0.80 0.75 0.70

3006 8981 26,181 62,725 1,061,610

1.3000 1.2205 1.0532 1.1352 1.0505

0.97 0.99 0.86 0.97 0.99

2268 6651 18,487 45,417 748,932

Weibull plot. The fitted results, as well as the coefficient of correlation r, are listed in Table 2. By setting the failure probability to 0.50 (i.e., PF = 0.50), the calculated fatigue life NPF=0.5 can also be obtained. This can be regarded as the mean fatigue life. The correlation coefficients indicate that the fatigue lives at various stress levels follow the Weibull distribution in general. The scale parameter increases as the stress level decreases. This reflects the fact that the fatigue life of a UHTCC becomes larger at a lower stress level, and it seems that the stress level has a limited effect on the shape parameter kN. 4.2. P-S-N models In fatigue situations, the material performance can be characterized by an S-N curve, also known as a Wöhler curve. In the following models, a modified S-N relation would be introduced to converge to the initial distribution of the static values. Generally, the S-N relation of concrete materials can be given in either of the following forms [31,36]:

NðSÞ ¼ aSb

ð3Þ

NðSÞ ¼ 10cdS

ð4Þ

where N is the fatigue life, S is the stress level, and a and b, as well as c and d, are empirical constants. Taking the logarithm for both sides of Eqs. (3) and (4) gives

log N ¼ log a b log S

ð5Þ

log N ¼ c d S

ð6Þ

In fact, the fitting results of Eqs. (3) and (4), as well as Eqs. (5) and (6), are very close. On the basis of the mean fatigue lives (i.e., NPF=0.5) presented in Table 2, the values of a and b can be fitted. These values are 180.926 and 21.679, respectively. The corresponding coefficient of correlation r is 0.975. Similarly, the values

of c and d can be obtained: 13.791 and 11.744, respectively. The corresponding coefficient of correlation r is 0.968. The above fitted results are shown in Fig. 8. They indicate that both Eqs. (5) and (6) can be used to describe the S-N relation of a UHTCC. It should be pointed out that these two S-N relations are generally obtained based on the mean value or average value of fatigue lives at various stress levels. This means that these S-N relations reflect fatigue lives with a failure probability of 0.50. For concrete materials, the stress level is a determining factor in the fatigue life. When the stress level decreases, the fatigue life increases significantly [8,31,36]. This can also be observed in Fig. 8. This means that the initial variation of strength can lead to a pronounced variation in the fatigue life of a fatigue test at the same stress level. In practice, it is essential to consider the reliability of the materials for structural design. Thus, probabilistic models of fatigue life based on the initial distribution of static strength and the S-N relation are developed in this study. The initial strength of the concrete materials may be introduced into the S-N relation using the following expression:

NðS; r0 Þ ¼ ðr0 =r0av e Þk N ðSÞ

ð7Þ

where r0 is the initial strength of the materials, r0-ave is the average initial strength (i.e., the strength used to determine the fatigue stress in a test, which is 3.015 MPa for the UHTCC under tensile load; see Section 2), and k is a transfer coefficient whose meaning will be explained later. The reasons why this function is used are as follows. First, it can be found in Eq. (7) that the fatigue life at the same stress level increases with an increase of r0, which is in accordance with the experimental findings. Second, N(S) can be obtained from Eq. (7) by setting r0 equal to r0-ave. Third, the introduced power function in Eq. (7) [i.e., (r0/r0-ave)k] is inspired by Eq. (3). That is, for a particular maximum fatigue stress, an increase in the static strength of the specimen leads to a decrease in the actual stress level, which results in an exponential increase in the fatigue life. Finally, introducing Eq. (7) into the two-parameter Weibull distribution will not change the type of distribution. This means that Eq. (7) can make a connection between various two-parameter Weibull distributions. This will be confirmed later. By introducing Eq. (7) into the distribution of the static strength of Eq. (1), the expression of PF for various stress levels can be obtained as follows:

0 PF ðN; SÞ ¼ 1 exp@

N ðk0 =r0av e Þk NðSÞ

!k0 =k 1 A

ð8Þ

It should be noted that Eq. (8) is also a two-parameter Weibull distribution with a scale parameter kN = (r0/rave0)kN(S) and a shape parameter kN = k0/k. As mentioned above, introducing Eq. (7) into the two-parameter Weibull distribution does not change the type of distribution. Furthermore, it can be observed that the transfer coefficient k builds a connection between the two shape parameters k0 and kN. It should be remembered that the stress level has a limited effect on the shape parameter kN. Thus, in this study, the average value of k0/kN is considered to be the value of k, which is 8.072. On the basis of Eq. (8), the fatigue life for a given S and PF can be derived as Eq. (9). k=k0

NðS; PFÞ ¼ ðlnð1 PF ÞÞ

Fig. 8. Fatigue life (PF = 0.50) and fitted S-N relations.

ðk0 =r0av e Þk NðSÞ

ð9Þ

P-S-N models of UHTCCs under tensile fatigue loads can be obtained by introducing Eq. (3) or Eq. (4) into Eqs. (8) and (9). In the following, the P-S-N model using Eq. (3) as N(S) is a model based on Eq. (3), and the other is a model based on Eq. (4). In Fig. 9, the distribution of the fatigue life at different stress levels given by Eq. (8) and the measured data are plotted. For

356


Fig. 9. Distribution of fatigue life at various stress levels.

comparison, the results of the model based on Eq. (3) and those based on Eq. (4) are both plotted. It can be seen that their results coincide. In addition, it can be observed that the agreement between the measured data and model results is good, and the fatigue life increases when the stress level decreases. It should be pointed out that the reason why the fitting results seem to be

slightly ‘‘shifted” for the stress levels of 0.80, 0.75 and 0.7 in Fig. 9 is that the corresponding mean fatigue lives do not perfectly coincide with the fitted curves in Fig. 8. It can be seen in Fig. 8 that the mean fatigue lives of S = 0.80 and 0.75 are a little smaller than the fitted results, while the mean fatigue life of S = 0.70 is larger than the fitted curves.

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In Fig. 10, Eq. (9) with the failure probabilities set to 0.10, 0.50, and 0.90 are plotted, and the measured fatigue lives are also presented for comparison. The results of models based on Eqs. (3) and (4) are shown in Fig. 10(a) and (b), respectively. It can be observed that most of the fatigue lives are located in the area between the dashed line (PF = 0.10) and the dotted line (PF = 0.90). That is, the proposed P-S-N model attains a considerable agreement with the experimental results. This model can be applied in the following conditions. On the one hand, when the stress level and the fatigue load cycles are obtained, the failure probability can be estimated. On the other hand, if the failure probability is set, the failure life under a specific stress level can be predicted. In summary, to apply this P-S-N model, the distribution of the static strength and one group of fatigue lives at a particular stress level are required to obtain the values of the parameters. It should be pointed out that the error of the predictions may be influenced by the choice of the stress level, and the accuracy of prediction could be improved by increasing the number of groups of fatigue lives at different stress levels. It should also be noted that Eqs. (3) and (4) are used as N(S) in this study. In fact, other functions that describe the S-N relation can also be used as N(S) in the proposed models. Furthermore, the above method was developed based on the tensile fatigue lives of a UHTCC, but the authors propose that a similar process can be used for other concrete materials. Additionally, as mentioned in Section 3, the areas of ‘‘rough” and ‘‘smooth” regions are linked to the stress level S. For the proposed fatigue model, the stress level S is an important parameter. It means that the fatigue-induced characteristics of the failure sur-

face have a certain repercussion in the model. However, the fatigue-induced crack pattern (or fatigue deformation) does not have any repercussion in the fatigue model, since the P-S-N model mainly describes the relationship between fatigue life and strength. In the following study, a micromechanics-based investigation of the fatigue deformation behavior of UHTCC under different stress levels needs to be carried out for a better understanding of the tensile fatigue behavior. 4.3. Comparison of fatigue lives For comparison, several S-N relations of the UHTCC, plain concrete (PC), steel fiber-reinforced concrete (SFRC), and ultra-high performance steel fiber-reinforced concrete (UHP-SFRC) are listed in Table 3. Eq. (3) is used to describe the S-N relation for the UHTCC, since its correlation coefficient is higher than Eq. (4) (see in Fig. 8). The minimum stress in Table 3 is the minimum stress in the fatigue tests, and rmax is the maximum stress in the fatigue tests. It should be noted that the influence of the minimum stress is considered in the S-N relations of PC in Refs. [26] and [27], and SFRC in Ref. [39]. Thus, specific values are set for these three S-N relations to minimize the influence of the minimum stress when compared with other S-N relations. It should be pointed out that the minimum stress of the flexural tensile fatigue test in [27] is compressive stress, and thus the minimum stress in the listed SN relation is set to zero. The stress levels in fatigue tests of UHTCC, PC, and SFRC in Table 3 were determined as the ratio of the maximum stress to the corresponding tensile strength. However, the stress level in the fatigue tests of UHP-SFRC in Ref. [40] was determined as the ratio of the maximum stress to the elastic limit

Fig. 10. Comparison of fatigue life and the P-S-N models: (a) model based on Eq. (3); (b) model based on Eq. (4).

Table 3 S-N relations of UHTCC and other concrete materials under tensile fatigue loadings. Materials

Fatigue loading

Fiber volume/%

Minimum stress

S-N relations

UHTCC in current research PC in Ref. [37] PC in Ref. [26] PC in Ref. [27] PC in Ref. [38] PC in Ref. [38] SFRC in Ref. [39] SFRC in Ref. [28] SFRC in Ref. [28] UHP-SFRC in Ref. [40]

Tensile Tensile Tensile Flexural tensile Splitting tensile Flexural tensile Tensile Tensile Tensile Tensile

2.0 0 0 0 0 0 1.0 0.75 1.5 3.0

0.1rmax 0.08r0-ave 0.1r0-ave 0 0.1rmax 0.1rmax 0.1rmax 0.29 MPa 0.29 MPa 0.1rmax

logS = 0.104–0.046 logN S = 0.987–0.041 logN S = 1.039–0.069 logN S = 1.144–0.091 logN S = 1.151–0.082 logN S = 1.143–0.079 logN S = 1.017–0.050 logN S = 0.985–0.046 logN S = 1.017–0.056 logN S = 1.199–0.088 logN

Note: PC = plain concrete; SFRC = steel fiber-reinforced concrete; UHP-SFRC = ultra-high performance steel fiber-reinforced concrete.

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On the basis of the initial distribution of static strength, novel P-SN models of UHTCCs under tensile fatigue loads are proposed. On the basis of the fatigue failure mechanism of UHTCC, the authors suggest that using some high-strength high-modulus fibers to replace partial PVA fibers in a UHTCC may achieve a longer fatigue life at both high and low fatigue stress levels. Conflict of interest None. Acknowledgements

Fig. 11. Comparison of tensile fatigue lives of UHTCC and other concrete materials.

strength of UHP-SFRC, which means the actual stress level is lower than that determined by the ultimate tensile strength. Thus, the ratio of the elastic limit strength (9.1 MPa) and the ultimate tensile strength (10.9 MPa) shown in Figure 1 of Ref. [40] was used to adjust the S-N relation, and the S-N relation of UHP-SFRC shown in Table 3 is the adjusted value. The S-N relations listed in Table 3 and the original test data for the UHTCC are plotted in Fig. 11. It can be observed that the S-N relation is close to that of UHP-SFRC in Ref. [28]. At a high stress level (i.e., S = 0.90, 0.85, and 0.80), the tensile fatigue life of the UHTCC is longer than those of plain and steel fiber-reinforced concrete in general. At a low stress level (i.e., S = 0.75, 0.70), the tensile fatigue life of SFRC seems to be longer. A similar phenomenon can be observed in the S-N relations of UHTCC under flexural fatigue loading [25]. This phenomenon may relate to the fatigue-induced rupture of PVA fibers in Section 3.4. As mentioned in Section 3.4, using fibers with a higher strength and modulus may improve the tensile fatigue behavior of UHTCC. In the flexural fatigue tests of Ref. [25], it was found that the fatigue lives of UHTCC using polyethylene (PE) fibers (tensile strength of 2790 MPa) are larger than those of UHTCC using PVA fibers (tensile strength of 1600 MPa) at low stress levels. This may be evidence for the above suggestion. Thus, the authors suggest that using some high-strength highmodulus fibers to replace partial PVA fibers in UHTCC may achieve a longer fatigue life at both high and low fatigue stress levels when compared to conventional concrete. Further investigation of this tentative idea is required. 5. Conclusion In this study, the tensile fatigue behavior of UHTCCs under various stress levels were investigated, and the following conclusions can be drawn. The number of cracks of UHTCC specimens under tensile fatigue loads decreases as the number of cycle increases. Four stages were observed in the evolution curve of fatigue deformation, which is different from the three-stage curve of conventional concrete. ‘‘Smooth” and ‘‘rough” areas can be distinguished on the fatigue failure surfaces of the UHTCC. The ‘‘rough” areas have large amounts of pulled-out PVA fibers. The ‘‘smooth” areas are formed during the propagation of fatigue-induced cracks, in which most PVA fibers are ruptured with very short ends out of the matrix. It is also found that the ‘‘smooth” areas become larger as the stress level decreases. The Weibull distribution is applied to describe the distribution of the tensile strength and fatigue life of a UHTCC.

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