Effect of Fiber Orientation on the Fatigue Behavior of Steel Fiber-Reinforced Concrete Specimens by Performing Wedge Splitting Tests and Computed Tomography Scanning

This paper shows the relationship, in steel fiber‑reinforced concrete, between fiber orientation and fatigue response through the combined use of computed tomography (CT), digital image processing (DIP) software and wedge splitting test (WST). The WST cubes were extracted from conventional 150 × 150× 600 concrete prisms and a groove and notch were carved on different faces in such a way that in half of the test specimens the fibers are oriented mostly perpendicular to the breaking surface and, in the other half, the fibers are mostly oriented parallel to the break‑ ing surface. Fiber orientation was obtained using a CT device and DIP software from a miniprism extracted from the previously mentioned concrete prisms. The results show that there is a strong correlation between the crack‑ sewing fiber orientation on the one hand and fatigue life and crack opening rate per cycle on the other hand. Cubes with a higher percentage of fibers perpendicular to the crack surface (i.e., with a higher efficiency index) show a longer fatigue life and a lower crack opening rate per cycle, while cubes with a lower efficiency index show a shorter fatigue life and a higher crack opening rate per cycle.


Introduction
Fiber-reinforced concrete is a technical solution of great interest in civil engineering due to the numerous advantages it offers over traditional structural concrete solutions.The fields of application of this building material are multiple, covering almost all structural elements traditionally built with reinforced concrete, as well as other new applications.Moreover, nowadays, due to the growing concern about the environmental impact of civil constructions, interest in fiber-reinforced concrete has increased and quite a few promising research projects are being carried out using natural fibers (Ahmand & Zhou, 2022;Khan et al., 2020;Shah et al., 2022).
One of the most common uses of fibers is to reduce concrete cracking, caused by shrinkage and thermal variations, among others.Therefore, fibers improve the durability of concrete (Nogueira et al., 2021;Saradar et al., 2018;Shen et al., 2022;Vrijdaghs et al., 2018;Zia & Ali, 2017).This application is particularly suitable for elements that do not require rebar structural reinforcement.In these situations, the minimum reinforcement ratios specified in the standards can be easily covered with normal fiber contents.
However, the greatest research efforts are being made in the field of the use of fibers as a structural component to replace the traditional rebar reinforcement totally or partially (Gao et al., 2021;Lee, 2017;Mobasher et al., 2014;Mudadu et al., 2018;Nilforoush et al., 2017;Ortega et al., 2022;Yoo & Yoon, 2015;Yuan & Jian, 2021).At this point, fiber-reinforced concrete is a very competitive option in civil engineering works, since it represents a significant cost saving, especially in labor, compared to conventional reinforced concrete.However, its implementation remains a technical challenge that has not yet been fully solved.
Among the different mechanical solicitations to which a fiber-reinforced concrete element may be subjected, fatigue is the one that currently arouses most scientific interest.There are two major structural elements where fatigue is the most important loading condition: concrete pavements and wind turbine structures (mainly foundations and towers).In both cases, the use of fiberreinforced concrete (instead of conventional reinforced concrete) has great technological interest (Ali et al., 2020a(Ali et al., , 2020b;;Batrukin et al., 2021;Cui et al., 2023;González et al., 2014;Ozturk & Ozyurt, 2022;Yazdanbakhsh et al., 2018).
In addition to the fiber content, the orientation of the fibers within the concrete matrix is an aspect that largely conditions the macroscopic response of the elements.However, its study is not straightforward, since the fiber arrangement is in principle random and cannot be easily controlled.It strongly depends on the size and shape of the specimens (due to the wall effect), the manufacturing process (whether the specimen is molded, drilled or cut) and the length of the fibers, among other aspects (Conforti et al., 2021;Laranjeira et al., 2011;Riedel & Leutbecher, 2021;Vicente et al., 2019a;Zhou & Uchida, 2017).In most of the published investigations, specimens showing uniform, preferred fiber orientations are usually tested.
The technical literature contains some interesting works in which series of specimens with clearly different fiber orientations have been fabricated.In this sense, it is worth mentioning the work carried out by Minguez and collaborators (Mínguez et al., 2018).In this paper, a small concrete wall was built from which concrete prisms were extracted according to different orientations and positions.These prisms were subjected to 3-point bending tests to evaluate the flexural strength.It was expected (as it finally happened), that the fibers would be aligned in the horizontal plane and substantially parallel to the main direction of the wall.Therefore, the prisms extracted in the horizontal direction showed fibers oriented mainly orthogonal to the failure surface, while in the prisms extracted in the vertical direction, the fibers were mainly oriented parallel to it.In addition, the fibers (steel fibers were used) tended to be located towards the bottom of the wall, so the percentage of fibers was higher in the prisms extracted from the bottom of the wall than in those from the top.
Similarly, the paper of Gonzalez and coworkers (González et al., 2018a) is noteworthy.In it, fiber-reinforced concrete prisms were fabricated from which cubes were extracted and then subjected to wedge splitting tests (WST) in two different configurations.In this case, it was expected that the fibers would be oriented in the horizontal plane, so that in one series of cubes the fibers would be arranged perpendicular to the cracking surface, while in another series the fibers would be mainly parallel to it.
However, in these two works fatigue tests were not performed, but static ones.In general, research focusing on the effect of fiber orientation on the fatigue response of concrete is scarce, being more common in compressive fatigue (Cachim et al., 2002;Cui et al., 2021;Khan et al., 2022;Medeiros et al., 2015;Mínguez et al., 2019) than in tensile (Deng et al., 2021;Isojeh et al., 2017;Martinello et al., 2019;Vicente et al., 2019b).
The main issue in any study of the influence of fiber orientation on the macroscopic behavior of fiber-reinforced concrete is to determine the real orientation of the fibers within the specimens.There are techniques that provide average values of fiber orientations based on indirect parameters such as inductivity (Al-Mattarneh, 2014;Li et al., 2020) or permittivity (Akgol et al., 2019;Mehdipour et al., 2017).These methods are fast, relatively inexpensive and can be used on site.However, their main drawback is their lack of accuracy.
X-ray micro-computed tomography is the technology that provides the most accurate results of the position, distribution and orientation of fibers in concrete.It requires expensive scientific equipment, but the results are of high quality and offer unlimited possibilities for research.In an X-ray tomography, a three-dimensional image of the sample is obtained, in which all the fibers inside the concrete are perfectly identified.
The research presented here is a continuation of González et al., 2018a.The fatigue wedge splitting tests performed here have the same setup as the static ones described in the previous work.
In this study, a fiber-reinforced concrete mixture was evaluated.Eight prisms of 150 × 150×600 mm were fabricated.Two 150-mm side cubes were extracted from each prism.The cubes were cut so that the fibers of each cube had a significantly different orientation.The orientation of the fibers of each prism was obtained by micro-computed tomography.Finally, the results were compared and analyzed.
This paper is structured as follows: The experimental procedure is presented in Sect.2, the results of the tests are described and discussed in Sect.3; and finally the conclusions are found in Sect. 4.

Experimental Program
The materials, the manufacturing procedure and the scanning procedure are described in this section.

Materials
Eight prisms 150 × 150 × 600 mm were cast to perform this work.Table 1 shows the mix proportion.
In addition, three cylinders 150 mm in diameter and 300 mm in height were performed and tested under compression to characterize their compression strength [according to EN 196-1:2016(British Standards Institution, 2016)].The specimens were cured in a humid chamber (20ºC and 100% humidity) for 28 days.They were then cured outside the humid chamber, in the laboratory, until the time of testing.The tests started when the specimens were 4 years old.At the beginning to the testing campaign, the average concrete compressive strength was 86.9 MPa with a standard deviation of 2.7 MPa.

Specimen Manufacturing Process
Eight couples of cubes (i.e., sixteen cubes) 150 mm in edge where extracted from the eight 150 × 150x600 mm prisms.Two cubes were obtained from each prism, located approximately at 1/3 and 2/3 of the lateral side of the prism (Fig. 1).Additionally, eight miniprims were extracted from the center of each prism for the CT scanning.
A groove and a notch were then carved in each cube (Fig. 2).On one of the couple's cubes, the groove and the notch were performed on the top face along the transverse axis of the prism.On the other couple's cubes, the groove and the notch were cut on the side face closest to the center of the prims and, again, along the transverse axis of the prism.The first set of cubes is called "top" while the second one is called "side".Finally, a total of eight top cubes and eight side cubes were performed.
It is expected that the fibers are mainly oriented parallels to the longest side of the prism.This means that, in the case of the top cubes, fibers are expected to be

Steel fibers 150
Fig. 1 Scheme of the extraction of the cubes and the miniprism oriented perpendicular to the notch and, finally, to the crack, while in the case of the side cubes, fibers are expected to be mainly oriented parallel to the notch and the crack.This forces the fibers to have clearly different orientations with respect to the plane of the crack.
In addition, a prismatic piece was cut from the center of each prism (Figs. 1 and 2).This piece as a squared section 0 mm side and a length of 150 mm.Because of the limitation of the CT-Scan device, these specimens were scanned to obtain the fibers orientation, instead of the whole prisms.Consequently, it is assumed that this miniprism is representative of the complete prism and, consequently, of the two cubes extracted from it (in terms of fiber orientation).

Scanning Process
The small prims mentioned above were scanned using a MicroCT Phoenix V|Tome|X s 240 unit from GE Sensing & Inspections Technologies (Boston, MA, USA), equipped with a high-power nanofocus X-ray tube (180 kV/20 W) and a high-power microfocus tube (240 kV/320 W).A total of 4,300 2D slices (Fig. 3) were obtained throughout the height of the miniprism.The voxel size is 35 × 35x35 μm 3 .Using the post-processing software package, a 3D image of the specimens was created.A gray level was assigned to each voxel of this tridimensional object depending on the real density of the matter at that point.Since 16-bit images are used, the gray scale varied from 0 to 65,535, where 0 means pure black, which is least dense (corresponding to pores) and 65,535 means pure white, which is densest (corresponding to steel fibers).The result of this first postprocess was a file including the X, Y and Z coordinates of the center of each voxel and its gray level.The total number of voxels in each miniprism was approximately 9.1•10 9 .The information of the files mentioned above was next post-processed using the commercial software AVIZO (FEI Visualization Sciences Group, Hillsboro, OR, USA).This tool identifies and isolates each individual fiber inside the prism.The first step consists on extracting all the voxels belonging to the fibers, which were defined on the basis of a certain gray level.In this case, a lower gray threshold of 30,000 was considered appropriate.Therefore, all voxels whose gray level was between 27,000 and 65,535 were considered as fibers.As a general criterion, the gray threshold is taken in a valley of the gray distribution histogram, since the valleys correspond to the boundaries between the different components of the material (Fig. 4).
Next, all the voxels in contact were added, since they belonged to the same fiber.Subsequently, the software proceeded to separate all those fibers that had been joined by small necks.In these cases, the voxel aggregation subroutine resulted in one fiber when in fact there are two or more fibers very close to each other.An algorithm based on watershed segmentation (Vicente et al., 2021) was applied.The final result of this image analysis is a new file containing the X, Y and Z coordinates of the center of each fiber, its length (as the largest Feret's diameter) and the orientation of the axis containing the length.The orientations is provided in Cartesian coordinates, i.e., the direction angles of the axis are given.Fig. 5 shows the 3D images of two of the six miniprisms.Fig. 5 shows how the fiber orientation is clearly different in two miniprisms extracted from two apparently identical prisms.The specimen on the left shows a lower fiber density.On the other hand, in the specimen on the right, the fiber density is clearly higher.Both miniprims show different fiber orientation patterns.
In this work, fibers with a length less than 2 mm were discarded as too small.These are ends or fibers, cut during the extraction process.

Wedge Splitting Test
The traditional test to measure the mechanical behavior of concrete under tension is the 3-point bending test (3PBT), which can be carried out according the standards EN 14651:2005 + A1:2007 (British Standards Institution, 2008) or ASTM C1609 (American Society for Testing & Materials, 2019), among others.One of the main limitations of this test is that the fibers tend to show a preferential orientation, due to the boundary conditions of the mold, the concreting direction, etc., which directly affect the results.
An alternative test to the 3PBT is the wedge splitting test (WST).This test was first proposed and described by Linsbauer and Tschegg (Linsbauer & Tschegg, 1986) and next developed by Brühwiler and Wittmann (Brühwiler & Wittmann, 1990).In this research, it is used to evaluate the fatigue life of fiber-reinforced concrete in tension.
Once the cubes had been extracted from the prism, they were prepared by sawing a groove and a notch (Fig. 6).As explained before, a novelty of this research is that the groove and the notch were sawed in two different faces, belonging to two different orientation.In consequence, it is expected that the orientation of the fibers is clearly different in the cubes belonging to the different series.
The cubes were placed on a square steel bar 20 mm in width, centered in the specimen and parallel to the groove's direction.Two steel L-shape devices equipped with two rollers each were placed on the top of the specimen at the both lips of the groove.In addition, a steel U-shape piece made up of two steel wedges were hanged from the hydraulic cylinder (Fig. 7).
During the test, the following parameters were measured: the applied load and the crack mouth opening displacement (CMOD).A tension-compression MTS 244.21 dynamic hydraulic cylinder (MTS, Eden Prairie, MN) was used, with a capacity of ± 50 kN.The actuator had a load cell MTS 661.20 F-02 (MTS, Eden Prairie, MN), with a range of ± 50 kN too and an error of below 1% of the range.For the measure of the CMOD, four laser distance sensor WENGLOR CP08MHT80 (WENGLOR, Tettnang, Germany) were used with a range of 50 mm and an accuracy of 0.01 mm.For the measure of the crack opening displacement, one MTS 624.12 F-24 clip gauge (MTS, Eden Prairie, MN) was used with a range of 15 mm and an accuracy of 0.01 mm.
The laser distance sensors were placed at the bottom of the notch, i.e., at the beginning of the crack to measure accurately the CMOD while the clip gauge was placed at the beginning of the notch, centered on the concrete cube (Fig. 7).
Two of the laser distance sensors were placed on the front face and the other two were placed on the rear face.The body of the sensor remained fixed and it monitor the distance to a L-shaped small piece glued to the specimen close to the end of the notch, i.e., at the beginning of the crack.The crack opening at a face was obtained as the sum of the displacement measured by the two laser distance sensors.Finally, the crack width, or CMOD, is defined as the average value of the measures of the crack openings of the front and rear faces.The measurement provided by the clip gauge was used as a contrast, to verify that the data provided by the laser distance sensors were consistent.
The test campaign had two stages.In the first stage, all the specimens were subjected to a static test until the crack appeared.During this test, an increasing vertical load F V was applied.The test was displacement- controlled and the tension speed was 0.05 mm/min until the maximum load is reached.The test stopped when the load decreased below 95% of the maximum vertical load or the CMOD reached a value of 0.5 mm (the condition that was first fulfilled).
At the end of this first stage of the test campaign, all the test specimens are pre-cracked.Then, in a second stage of the test campaign, the specimens were subjected to cyclic loading between 85 and 5% of the maximum load, measured in the pre-cracking test described above.The testing frequency was 5 Hz and the data recording frequency was 50 Hz.
In the wedge splitting test, the vertical force is decomposed, at the roller level in two forces, a vertical force 0.5 • F V and a horizontal force F H , according to the fol- lowing expression (Eq.1): where α is the wedge angle.In this case, α = 15º.
According to the Strength of Materials, both vertical and horizontal force cause a tension force and a bending moment at the collapse cross-section, according to the following expressions (Eq.2): The tension stress σ can then obtained considering a linear-elastic behavior of the material (Eq.3).

Results and Discussion
Several correlations between the fiber orientation and the fatigue behavior of fiber-reinforced concrete under tension can be obtained using the information provided by the CT-Scan and the one obtained through the wedge splitting test.

Fiber Orientation
The CT-Scan and the DIP software provide the necessary information to obtain the fiber orientation with respect to each of the Cartesian axes X, Y and Z (see Fig. 2), i.e., the direction angles of the length axis of the fiber.Next, the (1) ( histograms of angle distribution are shown for each miniprism (Figs. 9, 10, 11).Figs. 9, 10, 11 reveal that fiber are substantially oriented along the XY plane, i.e., perpendicularly to the Z-axis.This is the expected result, since the process of filling the prisms and the vibration of the molds means that the fibers tend to be arranged horizontally.With respect to the X and Y axes it is not observed a dominant orientation, i.e., fibers are uniformly distributed in the XY plane.It should be noted that, in the X and Y directions, some specimens show a slightly different behavior than the rest, presumably of a random nature.
Using the data shown in Figs. 9, 10, 11, it is possible to obtain a parameter called efficiency index with respect to the X, Y and Z axes, respectively ( ei x , ei y and ei z ), accord- ing to the following equations (Eq.4) (González et al., 2018a;Mínguez et al., 2018;Vicente et al., 2019a): where f i,x , f i,y and f i,z are the relative frequencies of the fiber orientation with respect to the X, Y and Z axes, respectively, α x , α y and α z the average angle of each range and n the number of ranges in which the histogram is divided (18 in this case).
The efficiency index may vary between 0 and 1, where 0 means that all the fibers as perpendicular to that axis and 1 means that all the fibers are aligned to that axis.
Table 2 shows the efficiency index values for all the prisms.In this research, it is assumed that the orientations of the fibers obtained from the miniprism are representative of the whole prism and, by extension, of the two cubes extracted from it.In this case, the data of interest is the efficiency index with respect to the axis perpendicular to the crack plane.In the case of top cubes, the efficiency index to be considered is ei x , while in the case of side cubes, the efficiency index of interest is ei z .

Pre-cracking Test
Table 3 shows the maximum vertical load P max and the ultimate vertical load P ult (i.e., the load when the test ends) obtained in the pre-cracking test for each of the specimen.
In the specimen identification code, C stands for "cube", the number represents the prism from which the cube is extracted and the letter at the end represents   the position of the groove.In this case, "T" stands for "top" while "S" stands for "side".The results show that the cubes belonging to the T-position have a slightly higher maximum load than those belonging to the S-position.Specifically, the average value of the maximum load is 7.43 kN for the T-position and 6.76 kN for the S-position, which is approximately 9.8% more.When comparing the ultimate load values, the results show the same behavior, but more pronounced.In this case, the average value of the maximum load is 7.26 kN for the T-position and 6.52 kN for the S-position, which represents approximately 11.4% more.This shows that the S-position cubes, in which the fibers have an orientation substantially parallel to the notch and the crack, show a slightly less ductile behavior than the T-position cubes, in which the fibers have an orientation substantially perpendicular to the notch and the crack.
The fact that the cubes belonging to the T-position show a higher maximum load than those at position S denotes that the fibers cause an increase of the tensile strength of the concrete in their direction, which agrees with previous work performed by the authors (González et al., 2018a).

Fatigue Test
After the pre-cracking tests, all specimens were subjected to a cyclic load test up to specimen rupture.The maximum and minimum load values are given in Table 3.All the testing parameters have been described before.It is important to highlight two issues.First, given that the wedge splitting test can only apply tensile stresses to the specimen, the test performed is a tensile-tensile fatigue test.Second, this test does not evaluate the fatigue life of the concrete matrix in tension, but that of the reinforcement.That is, because during the first stage of the test, the concrete cracked and, therefore, does not directly contribute to the fatigue life.Then, there are two main mechanisms of the fibers that are capable of withstanding fatigue stresses: the tensile strength of the steel itself and the bonding forces of the fibers to the concrete matrix.
Commonly, fatigue tests are performed without precracking.In this case, during the first part of the fatigue test, concrete bears the cyclic loads until it cracks.From this point on, fibers withstand the cyclic loads.In the case of fatigue test with pre-cracking (González et al., 2014(González et al., , 2018b;;Martinello et al., 2019;Vicente et al., 2019b), concrete matrix does not cooperate directly in supporting cyclic tension caused by the load cycles, because it is pre-cracked.The result is a smaller fatigue life.In any case, the concrete matrix indirectly cooperates in the fatigue response of the specimen by controlling the bonding capacity of the fibers and their degradation mechanisms, and consequently, their ability to withstand cyclic loading.
Table 4 shows the fatigue life of each specimen.Fig. 12 shows the fatigue life results of the different specimens.Additionally, a Weibull fitting of the fatigue life results of the cubes belonging to the T-position on the one hand and those belonging to the S-position on the other hand has been performed.These curves are also shown in Fig. 12.
The results show how the cubes belonging to the T-position have a longer fatigue life than those belonging to the S-position.This value can be estimated to be approximately 5 times.
In the cubes belonging to the T-position, the fibers are better oriented (i.e., more clearly oriented in the direction of the tensile stresses), so they are more efficient.This results in a longer fatigue life.On the contrary, in the cubes belonging to the S-position, the fibers are mainly oriented in a direction perpendicular to the tensile stresses, so they are less efficient and, consequently, the fatigue life of the specimen is lower.
The fatigue behavior of the cubes can be further studied through the evolution of the crack opening with the loading cycles.The fatigue creep curves are the envelope of the maximum crack opening, which are shown in Fig. 13.Additionally, Fig. 14 shows the normalized fatigue creep curves, where abscissa is the damage "D", which is defined as the ratio between the number of cycle "n" and the fatigue life "N".Thus, all curves go from 0 to 1. Fig. 13 shows how all the fatigue creep curves show very similar behavior.The curves can be divided into three parts.During the first part, there is a very rapid increase in crack opening.This increase gradually dampens until the second part is reached, where the growth is approximately linear.At the end, during the third part, a very rapid increase of the crack opening is observed again until the specimen breaks.This phenomenon has been widely observed in compressive fatigue tests (Medeiros et al., 2015;Ortega et al., 2022) and also in flexural fatigue tests (Martinello et al., 2019).The results obtained in this work (and shown in Fig. 14) demonstrate that this behavior also occurs in wedge splitting test where the CMOD is measured.This behavior is expected to be characteristic of concrete elements, regardless of the type of test performed.In this case, the first part of the curve represents approximately the first 1% of the curve, while the third part represents approximately the 10% of the curve.These percentages are slightly different from those observed in the case of compressive fatigue.In particular, When the specimens are cracked, all this phenomenon has almost completely occurred during the pre-cracking process, so the first part of the curve is much smaller.
From the second part of the fatigue creep curve its slope, dδ/dn , can be determined.By analogy with compressive fatigue, this value is determined as crack opening rate per cycle (Fig. 15).
Table 5 shows the value of this parameter for all the specimens.

Correlation Between Fatigue Life and Crack Opening Rate Per Cycle
The crack opening rate per cycle has a linear relationship with the fatigue life N on a bi-logarithmic scale.This relationship was first shown in Sparks and Menzies (1973) for compressive fatigue testing and is commonly known as the Sparks-Menzies relationship.Fig. 16 shows the correlation between crack opening rate per cycle and fatigue life for all cubes, just with the linear fit to the data points.As can be seen in Fig. 16, the results of all the cubes together show a good correlation (coefficient R 2 = 0.98), which reveals that the relationship between crack opening rate per cycle and fatigue life is independent of the fibers; it only depends on the concrete matrix.The specimens with the best oriented fibers (T-Position Cubes) show a higher fatigue life and a smaller value of the crack opening rate per cycle, i.e., these specimens are located in the lower right half of the graph.In contrast, the specimens with the worst oriented fibers (S-Position Cubes) show a shorter fatigue life and a higher value of the crack opening rate per cycle, i.e., these specimens are located in the upper left half of the graph.

Correlation Between Fatigue Life and Efficiency Index
Fatigue life is related to fiber content and fiber orientation, as the fibers help to withstand the cyclic loads and, therefore, the more fibers and/or the better the fiber orientation, the longer the fatigue life.
In this particular case, all the cubes have approximately the same fiber content, but the orientation of the fibers is significantly different between the cubes belonging to the T series and the cubes belonging to the S series (this is an advantage of this test compared to the conventional three-point bending test).As explained above, fibers are approximately oriented in the horizontal plane of the prism, which means that they are oriented sensibly perpendicular to the crack in the case of the specimens belonging to the T series, and they are oriented sensibly parallel to the crack in the case of the specimens belonging to the S series.Fig. 17 shows the relationship between the fatigue life and the efficiency index (ei x for T series specimens and ei z for S series specimens).
The results show that there is a correlation between both parameters, such that the higher the efficiency index, the longer the fatigue life.This conclusion can also be drawn from Tables 2 and 4.These tables, analyzed together, show that the T series has higher fatigue life values and higher efficiency index, while the S series has lower fatigue life values and lower efficiency index.
However, the correlation is not so strong than one might think a priori (coefficient R 2 = 0.58).This is because the efficiency indices have been obtained, for each prism, from its corresponding miniprism (see Fig. 2) and have been considered the entire miniprism.
At this point, when Fig. 5 is observed in detail, it can be detected that the fibers are clearly oriented in the XY plane, especially in the lower and upper parts of the concrete prism (and, therefore, also in the miniprism), which are two of the regions affected by the "wall effect".However, in the central part of the miniprism, the percentage of fibers with a more vertical orientation is higher.If a local analysis is carried out, focused only on the fibers that pass through the cracks during the test, it can be observed that the fiber orientation is slightly different.In the case of cubes belonging to the T series, the fibers that govern the fatigue response are those located in the lower half of the cube, with those located in the part closest to the middle plane of the cube being more relevant.In the case of the cubes belonging S series specimens, the fibers that govern the fatigue response are those located in the middle plane of the cube.Extrapolating this Extrapolating this reasoning to the concrete prism (or miniprism), it is concluded that the fibers located in the central region of the prism have the greatest influence on the fatigue response of the cubes.
Table 6 shows the efficiency indexes ei x and ei z for all the prisms particularized for the central third.
Fig. 18 shows the relationship between the fatigue life and the efficiency index of the central third (ei x for T series specimens and ei z for S series specimens).
In this case, the correlation is now strong (R 2 = 0.82).

Correlation Between Crack Opening Rate Per Cycle and Efficiency Index
Fig. 19 shows the relationship between the crack opening rate per cycle and the efficiency index.Again, there is a relationship between the two parameters, so that the higher the efficiency index, the lower the crack opening rate per cycle.This means that the better the fiber orientation, the smaller the crack growth per cycle.
As in the previous case, the relationship between the two parameters is not very strong (coefficient R 2 = 0.57).The explanation is the same as the one described above; the efficiency index used in the comparisons is the one obtained from the miniprism, implicitly assuming that it is the same throughout the prism (and, therefore, also in the two cubes coming out of it).However, when considering the fibers that cross the crack, which in this case corresponds approximately to the fibers located in the central third of the miniprism, the correlation improves substantially.
Fig. 20 shows the relationship between the crack opening rate per cycle and the efficiency index of the central third.

Conclusions
In this research work, the influence of fiber orientation on the fatigue response of steel fiber-reinforced concrete is analyzed.The evaluation of the fiber arrangement has been carried out by combining micro-CT scanning and post-processing with image analysis software.As for the fatigue response, wedge splitting tests (WST) were performed.The concrete mix produced had a fiber content of 2% by volume.A total of 8 standard prisms of 150 × 150×600 mm were fabricated.Then, from each prism, 2 cubes of 150 mm edge and a miniprism of 50 × 50 × 150 mm were cut.The cubes were next grooved and notched.In one cube per prism, the groove and notch were made on its top face, while in the other cube, they were made on its side face.In this way, it is ensured that the preferred orientation of the fibers (with respect to the crack surface) in each cube is significantly different.In any case, the actual fiber orientation of each prism was obtained with a micro-CT scanner.
Based on the findings of the experimental program presented in this paper, the following conclusions can be drawn: 1. Micro-CT reveals that the fibers tend to be positioned in the horizontal plane, i.e., perpendicular to the Z-axis, the one corresponding to the direction of concrete pouring.In the XY plane, no clear orientation tendency is observed.Next, the efficiency index is obtained for each prism and each direction (X, Y and Z).In all cases, the relevant efficiency index is the one belonging to the axis perpendicular to the cracking surface of the cubes.In the case of the T-position cubes, the efficiency index along the X-axis governs their mechanical behavior, while in the case of the S-position cubes, it is the efficiency index along the Z-axis that does so.
2. Pre-cracking static tests reveal that the T-position cubes have a slightly higher maximum load and ultimate load than the S-position cubes.This finding shows that the T-position cubes, whose fibers are better oriented, have a higher mechanical capacity. 3. Fatigue tests show that the T series cubes have, in general, a longer fatigue life than the S series cubes.
On the other hand, the analysis of the fatigue creep curve reveals that the first stage of the diagram is very small (only 1% of the total).This particularly low value, compared to other fatigue tests, is explained by the fact that the specimens are pre-cracked.On the contrary, the crack opening rate per cycle shows smaller values for the T series cubes than for the S series.4. When the fatigue life is plotted against the crack opening rate per cycle (both parameters in logarithmic scale), a strong linear relationship between them is observed.In this case, all the tests of both series are aligned, which shows that this correlation (Sparks and Menzies' Law) does not depend on the fiber orientation, but only on the concrete matrix.5.If the fiber efficiency indexes are compared with the fatigue parameters (fatigue life and crack opening rate per cycle), it is found that there is a clear relationship between them.The relationship is not very strong when considering the efficiency indexes over the entire volume of the miniprism.However, when the efficiency indexes of the central third of the miniprism (which has been assumed to be similar to that of the fibers crossing the cracks) are considered, the relationship is much more robust.

Fig. 2
Fig. 2 Geometry of the cubes and the miniprism

Fig. 5
Fig. 5 Examples of 3D view of the fibers' distribution in two miniprisms

Fig. 6 Fig. 7
Fig. 6 Dimensions of the groove and the notch

Fig. 8
Fig. 8 Mechanical behavior of the WST

Fig. 9
Fig. 9 Histogram of angle distribution with respect to the X-axis

Fig. 10
Fig. 10 Histogram of angle distribution with respect to the Y-axis

Fig. 11
Fig. 11 Histogram of angle distribution with respect to the Z-axis

Fig. 12
Fig. 12 Experimental fatigue life and Weibull fittings

Fig. 16
Fig. 16 Correlation between the crack opening rate per cycle and the fatigue life

Fig. 17
Fig. 17 Correlation between the fatigue life and the efficiency index

Fig. 18
Fig. 18 Correlation between the fatigue life and the efficiency index of the central third

Fig. 19
Fig. 19 Correlation between the crack opening rate per cycle and the efficiency index

Fig. 20
Fig. 20 Correlation between the crack opening rate per cycle and the efficiency index of the central third

Table 2
Efficiency index

Table 3
Results of the pre-cracking test

Table 4
Fatigue life

Table 5
Crack opening rate per cycle

Table 6
Efficiency index of the central third