1

INFLUENCE OF FIBER ORIENTATION ON THE BRIDGING

2

PERFORMANCE OF PVA FRCC

3

Toshiyuki Kanakubo, Masaru Miyaguchi and Kohei Asano

456

Biography: ACI member Toshiyuki Kanakubo is an Associate Professor at the Department

7of Engineering Mechanics and Energy, University of Tsukuba, Japan. He received his Ph.D.

8

from University of Tsukuba. His research interests include high-performance fiber-reinforced

9cementitious composites (HPFRCC), the structural behavior of FRP reinforced concrete

10

structures, and bond properties of reinforcement and concrete. He is the secretary of the

11

Technical Committee ISO/TC71/SC6, Non-Traditional Reinforcing Materials for Concrete

12Structures.

13

Masaru Miyaguchi is a student in the master’s program at the Department of Engineering

14

Mechanics and Energy, University of Tsukuba, Japan. He received his B.E. from University

15

of Tsukuba, Japan. His research topic is the evaluation of fiber orientation in high-16

performance fiber-reinforced cementitious composites (HPFRCC), and their influence on the17mechanical behavior of HPFRCC.

18

Kohei Asano is a Research Associate at the Department of Architecture, Miyakonojo

19College, National Institute of Technology, Japan. He received his Ph.D. from University of

20

Tsukuba, Japan. His research interests include the mechanical behavior of high-performance

21

fiber-reinforced cementitious composites (HPFRCC), and structural performance of RC

22members using HPFRCC.

23

1

1

ABSTRACT

2

Crack bridging performance of fibers strongly affects the tensile characteristics of fiber-

3

reinforced cementitios composites (FRCC) after first cracking. The fiber orientation

4

distribution is likely to be affected by factors that include fresh-state properties, casting

5

method, formwork geometry, and others. The objective of this study is to investigate the6influence of the fiber orientation on the bridging performance in polyvinyl alcohol (PVA)

7

FRCC through a visualization simulation using a water glass solution and a calculation of the8

bridging law. The main parameter of the investigations, in the present study, is the casting9

direction. To evaluate the fiber orientation distribution quantitatively, an approximation

10

methodology using an elliptic function is newly introduced. The bridging stress versus crack

11width relationship is calculated considering the elliptic distribution, the snubbing effect, and12

the fiber strength degradation. The calculated stress – crack width curves can express well the13

uniaxial tension test results after first cracking.

14
15

Keywords: FRCC; casting direction; fiber orientation; image analysis; bridging law; uniaxial

16

tension test; elliptic function; orientation intensity

17

2

1

INTRODUCTION2

The crack bridging performance of fibers, that is generally expressed by a3

bridging stress – crack opening relationship (called the bridging law), strongly affects the4tensile characteristics of fiber-reinforced cementitious composites (FRCC) after first cracking.

5

The bridging performance is characterized and/or controlled by the properties of the matrix,

6

the fiber, and the fiber-matrix interface 1,2. Since the 1980s, studies on high-performance

7

fiber-reinforced cement composites (HPFRCC) and engineered cementitious composites

8(ECC) have been conducted to understand the crack bridging performance, primarily because

9

these composites require the balanced properties of the matrix, the fiber and their interface, to10exhibit the pseudo strain-hardening behavior 3,4. One of the examples of a polymeric fiber11

bridging law is that presented by Kanda and Li 5 who described it for polyvinyl alcohol

12(PVA) fibers assuming the following characteristics: (1) the chemical bond in the fiber-

13

matrix interface, (2) the rupture of the fiber, and (3) the tensile strength reduction owing to14

inclined-angle bridging. These considerations had primarily been introduced to account for15

the characteristics of randomly oriented, discontinuous fibers 6.

16

Many researchers have studied the effects of fiber orientation on the mechanical17

characteristics of FRCC, including fiber-reinforced concrete (FRC). The categories of these18

materials including HPFRCC and ECC are summarized in some literatures e.g.7. In addition,

19

self-consolidating concrete (SCC) and ultra-high performance fiber-reinforced concrete

20(UHP-FRC) have been developed for the last several decades. These materials have specific21

properties that require researchers and engineers to be attentive to fiber orientation. The

22

scheme of the current approach to evaluate the fiber orientation has considered the casting

23

method, fresh-state properties, flow, vibration, and formwork geometry 8. The cementitious

24

matrix used in HPFRCC and ECC has a high viscosity aiding the random distribution of the25

fine fibers, and commonly has self-compacting properties. These characteristics indicate that3

1

the bridging law in HPFRCC and ECC is likely to be affected by the fiber orientation. In fact,

2

the tensile characteristics of polymeric fiber-reinforced cementitious composites differ

3

because of the casting direction and the dimension of the specimen 9. The wall effect, in

4

which the fiber orientation is influenced by the surface of the mold, has also been studied by5

many researchers. Li and Wang 10 categorized the fiber orientation as two-dimensional (2-D)

6

random and three-dimensional (3-D) random by the specimen dimensions in two

7

perpendicular sectional planes. The ultimate tensile strain of PVA-ECC tends to decrease if

8

the specimen dimension changes from 2-D to 3-D.

9

Statistical approaches on the fiber orientation distribution began in the 1960s.

10

Naaman 11 proposed a sinusoidal function as the probability density function (PDF) of the11

angle between the fiber and the normal vector of the cut plane. Stroeven 12 indicated the12

combination of three typical distributions, namely, 3-D random, 2-D random, and perfectly

13aligned one-dimensional for simulation of arbitrary orientation distributions. One of the14

examples of the approaches adopted to study the wall effect is presented by Dupont and15

Vandewalle 13. They proposed a theoretical quantification by predicting the total number of

16

fibers crossing a rectangular section. Considering the influence of the matrix flow, Xia and17

Mackie 14 proposed the probabilistic spatial orientation using the beta distribution as the18

axisymmetric fiber orientation. There have been lots of studies to investigate the fiber

19

orientation and distribution by experimental approaches observing fibers directly via image-20

based analysis. In case of steel fibers, X-ray technique is one of the effective methods.

21

Recently, micro-computed tomography (micro-CT) has been used to characterize the fiber

22

distribution 15. In case of polymeric fibers such as PVA, image analysis taking advantage of23

absorbing the ultraviolet radiation was conducted 16.

24

It is considered that the bridging performance, i.e. the tensile properties of FRCC,

25

can be characterized using the fiber orientation distribution. The main objective of this study4

1

is to investigate the influence of fiber orientation distribution on the bridging law of2polymeric fibers. The main experimental parameter selected in this study is the casting3

direction, which is considered to have an influence to the fiber orientation distribution.

4

To achieve the goal, a visualization simulation is conducted using sodium silicate

5solution (known as water glass) to observe the flow patterns of the fibers in the tension test

6

specimen. The results of the visualization simulation are discussed mainly for the distribution

7of the angles of the fibers. In this study, based on the visualization results, a new PDF is

8

proposed to describe variation in the fiber angle. The PDF is expressed by two parameters:9

the principal orientation angle and the orientation intensity. These parameters indicate the10angle and the tendency of the fibers to orient along the direction of the principal orientation.

11

Finally, the bridging law, which is obtained by a numerical calculation, is compared with the12tension test results.

13

The fine fibers with a diameter ranging between 0.01 mm – 0.04 mm (410-4 in. –14

1610-4 in.) are commonly used for HPFRCC/ECC to actualize the pseudo strain-hardening

15

behavior and multiple cracking. On the other hand, multiple cracking makes the observation

16of the bridging law difficult. In this study, PVA fiber with a diameter of 0.10 mm (3.910-3

17in.) is utilized to observe the bridging law (tensile stress – crack width curve) directly by the18

tension test subjected to single crack formation.

19
20

RESEARCH SIGNIFICANCE

21

Evaluation of the bridging law, accounting for the fiber orientation distribution, is22

necessary for predicting the precise tensile characteristics of FRCC. The fiber orientation,

23

which is affected by casting method, fresh-state properties, flow, and formwork geometry,24

should be considered in the manufacturing of the composites for practical uses. The bridging

25

characteristics of polymeric fibers are affected by their angle with the cracking plane.

51

Understanding the fiber behavior expressed by the bridging law can facilitate understanding2

the tensile characteristics of FRCC. A simple mathematical expression for the PDF of the3

fiber orientation distribution would also simplify simulations of the bridging law.

4
5
6

VISUALIZATION SIMULATION OF FIBER ORIENTATION
Materials for simulation test

7

PVA fibers of 0.10 mm (3.910-3 in.) diameter were utilized in this study. The

8mechanical properties of PVA fibers are listed in Table 1. In order to visualize the flow of the

9fiber in a matrix, a sodium silicate solution (hereafter referred to as water glass) was adopted

10

as the matrix. Water glass has high viscosity, and it is colorless and transparent. In regards to11

the practical use of ECC, the rheology of mortar matrix before mixing the fiber was inspected

12

using the flow time 17, based on “Test method for flowability of grout for prestressing

13

tendons (JSCE-F531-2013)” 18. The flow time is measured using the funnel shown in Fig. 1.

14

The flow time of water glass was controlled by adding pure water, in an effort to attain the15same flow time of mortar matrix as that of the target HPFRCC. The mix proportion of the16target HPFRCC is listed in Table 2. This proportion is selected for the tension test specimens,17

as explained in a later section. The measured flow time of the mortar matrix was 36 sec. in18

average for each of the eight mixture batches with the same mix proportion. The water glass

19

to the pure water weight ratio was chosen to be 12:1 at a temperature of 25 °C. The density of20

the water glass solution was 1.62 g/cm3 (101 lb/ft3), which is smaller than 1.89 g/cm3 (11821lb/ft3) of mortar matrix used for the tension test specimens.

22

The color of PVA fibers, which is almost white, makes it difficult to distinguish23

them from the water glass solution. Therefore, black-colored “target fibers” made from Nylon

24

were added to the matrix to simplify the image analysis. The mechanical properties of the25

target fibers are listed in Table 1. The volume fraction of the target fibers was set to 0.05%

6

1

based on empirical trial and error results. The mixture states are shown in Fig. 2. The image

2

analysis on target fibers (explained in a subsequent subsection) was conducted based on the3assumption that these fibers flow in similar orientations as those associated with the PVA

4fibers.

5

Simulation method

6

Water glass solution containing PVA and the target fibers was poured into the7mold, using the same way as that used for HPFRCC casting. The mold was constructed with

8transparent acrylic plates. For simulations of the flow in the tension test specimens, the cross-9

section of the mold was chosen to be 40 mm  40 mm (1.57 in.  1.57 in.) to be over three

10times the fiber length of 12 mm (0.47 in.), considering 3-D orientation of fibers 10. The

11

testing parameters included the casting direction and the volume fraction (Vf) of the PVA

12fibers. The dimensions of the mold are shown in Fig. 3. Two molds were prepared, one for13

the casting along the horizontal direction and a second for the casting along the vertical14

direction. Water glass solution was poured into the mold using a bucket at the points

15

indicated by the arrows in Fig. 3. The pouring time was approximately 20 sec. and was16

similar in value to the case of casting of the tension test specimens. After pouring, photos of17

the x-y and z-x planes were taken using two digital cameras at in-plane resolution of 6000 

184000 pixels. The setup of the cameras for the horizontal casting simulation is shown in Fig. 4.

19

Simulated volume fractions of PVA fibers are 0.1%, 0.5%, 1.0%, 1.5%, and 2.0%. For each20volume fraction, three image specimens were cast followed by photo capturing were carried

21

out. An example of the photograph (Vf = 0.1%, horizontal casting, z-x plane) is shown in Fig.

22

5.

23

Image analysis and calculation of fiber angle

24

Image analysis was conducted to obtain the fiber angles in the water glass

25

solution. The image analysis and calculation of the fiber angles were carried out for the target7

1

fibers that occupied the central 40 mm (1.57 in.) region as shown in Fig. 5. The procedure of2

the image analysis is described as follows:

3(1) The photograph is cropped to include only the target region. (Fig. 6 (a))4(2) The image is binarized and the noise is filtered. (Fig. 6 (b))5(3) RGB values (red-green-blue values in bit for each color) of the pixel data are read with

6

position coordinates (Xi, Yi).

7(4) The sequences of black-colored pixels are grouped and labelled. (Fig. 6 (c))

8After this process, a straight line approximation is calculated using the position coordinates of9

the pixels of the same group, using least-squares regression analysis by minimizing the10

distance between the point and the line. The fiber angle is defined as the angle between the11fitted line and the longitudinal axis (x-axis). The fiber angle ranges between  90 ° to  90 °.

12

Examples of fiber angle histograms (Vf = 0.1%, horizontal casting) are shown in

13

Fig. 7. The diagram on the right side of this figure corresponds to the calculated histogram

14

result based on the photograph of Fig. 5, and the analysis methodology shown in Fig. 6. As

15

indicated in Fig. 6 (b), fiber angles mostly range between 0 ° to 45 °. The frequencies of the16

fiber angles that are extracted based on the three-time pouring and photography are added

17

together, and one diagram is drawn for each parameter of the simulation test. All the fiber

18

angle histograms are shown in Fig. 8. As expected, there is a tendency that the fibers flow

19

along the longitudinal direction in the case of horizontal casting, and along the perpendicular

20

direction in the case of vertical casting. The presented solid lines and the respective values of21

the diagrams are explained in the next section.

22
23
24

PDF FOR FIBER ORIENTATION DISTRIBUTION
Approximation based on the elliptic function8

1

For the purpose of quantitative evaluation of the fiber orientation distribution, an

2

approximation methodology using the elliptic function is introduced. This methodology was

3studied in the field of “Japanese traditional paper (Washi)” 19. The relative frequency for each

4class of fiber angle is transformed into a vector with the argument set to be equal to the fiber

5

angle as shown in Fig. 9. The trajectory traced by the terminal points of these vectors is

6

approximated by an ellipse fitted using the least squares method. The ellipse is expressed as a7function of two radii, a and b, and the angle with respect to the x*-axis,  r , as shown in the

8figure. The value of  r ranges between  45 ° to  45 °, and the argument of radius a9corresponds to  r . As shown in Fig. 9, a random fiber orientation results in a circle whereas

10

the orientation tendency of the fibers along the longitudinal direction results in an ellipse. As

11the longitudinal directionality becomes greater, the shape of the ellipse becomes narrower.

12

The ratio of the two radii, k  a / b , can express the shape of the ellipse. This ratio of two

13

radii is defined as “orientation intensity”, and the angle,  r , is defined as “principal

14

orientation angle”. The orientation intensity value reflects the orientation tendency of the

15

fibers that lie along the principal orientation angle. When the fibers orient perfectly randomly,

16

k is equal to 1. As shown in Fig. 10, when the fibers show an increased directional

17

orientation toward the principal orientation angle, the value of k is larger than 1. In contrast,

18

when the fibers orient perpendicularly with respect to the principal orientation angle, the19value of k is smaller than 1.

20

The PDF that expresses the relative frequency corresponding to the fiber angle,  ,21

is described by Eq. (1): hereafter, the PDF is referred to as “elliptic distribution”. The

22

parameters for this function are the orientation intensity, k, and the principal orientation angle,23

 r . When  r is equal to zero, the elliptic function is simply given by Eq. (5). The definite

24

integral calculus of Eq. (1) and Eq. (5) in   / 2     / 2 gives 1 (the sum of probability).

9

k

C
cos   A sin  cos  B sin 2 1

p ( ) 

2A(1  k ) sin 2 r
1  (k  1) sin 2  r(2)

3k  (k  1) sin 2  rB
1  (k  1) sin 2  r

(3)

4

C

1
1  (k  1) sin 2  r(4)5

p ( ) 

6



k







2

1
cos   k  sin 2 

(1)

(5)

2

Approximation of visualization simulation results

7

The results of the approximation of the fiber angle distribution obtained in the8visualization simulation are shown in Fig. 8 by solid lines. The values of the orientation9

intensity, k , and the principal orientation angle,  r , are also listed in the figures. When the10

directionality of the fiber orientation increases along the longitudinal direction, the value of11

the orientation intensity is over five (cases of z-x plane for Vf = 1.0, 1.5 and 2.0%). In vertical

12

casting, the fiber angles tend to align along the perpendicular direction, and there are the13cases that the value of the orientation intensity becomes smaller than 0.5 (cases of x-y and z-x

14planes for Vf = 1.5 and 2.0%). These evaluations are done for two planes individually. The

15

estimated probabilities for each plane are multiplied to express the probability in 3-D

16orientation in a later section.

17
18
19

UNIAXIAL TENSION TEST
Test outline

20

For verification of the influence of fiber orientation on tensile behavior, the21uniaxial tension test was conducted. As explained in the Introduction, PVA fibers with a10

1diameter of 0.10 mm (3.910-3 in.) are utilized to observe the bridging law directly, subjected2

to single crack formation. The mechanical properties of the PVA fiber are listed in Table 1.

3

The fibers used for the tension test are same as those used in the visualization simulation. The

4mix proportion of mortar matrix has already been presented in Table 2. The fiber volume5

fraction is 2.0 %.

6

The testing parameter is the casting direction along both the horizontal and7vertical directions. Two types of molds for each casting direction were prepared, as shown in

8

Fig. 11. The matrix with fibers was poured into the mold using a bucket employing the same9

approach as the one used for the visualization simulation. The pouring time was controlled to10

be approximately 20 sec. in the test region.

11

The dimensions of the specimen and the specimen setup are shown in Fig. 12.

12

The cross-section of the test region is 50 mm  50 mm (1.97 in.  1.97 in.) square to be over13

three times the fiber length of 12 mm (0.47 in.), considering 3-D orientation of fibers 10. The

14

total length of the specimen is 510 mm (20.1 in.). A 2,000 kN (450 kips) universal loading

15

machine was used. Pin-fix ends were used at the boundaries to minimize possible effects of16

development of external moment because of setup irregularity, and secondary moment

17

influencing local fracture 9. The carbon fiber sheets were attached at both ends to avoid peel-18

off of the steel plate. Measurement items were tensile load and deformation in the test region

19

using two pi-type displacement transducers. Two series of test in different period (Batch #1,

20

compressive strength 39.2 N/mm2 (5.69 ksi), and Batch #2, compressive strength 41.0 N/mm2

21(5.95 ksi)) were carried out.

22

Test results

23

All specimens fractured by a single crack. Some of the specimens had a fine24

crack before loading because of an unskillful treatment during the formwork removal. The

25

test results of these specimens are excluded from the following discussions. The curves of the11

1

tensile stress – crack width are shown in Fig. 13. It is clearly recognized that the casting2

direction remarkably affects the tensile performance. The test results are summarized in

3

Table 3. The average tensile stress at the maximum load after the sudden drop of the load

4

(second peak) is 3.51 N/mm2 (0.509 ksi) and 1.67 N/mm2 (0.242 ksi) for the horizontal

5

casting and the vertical casting specimens, respectively. The tensile stress of the second peak6of horizontal casting specimens is more than two times that of the vertical casting specimens.

7

The crack width at the second peak of the horizontal casting specimens is on average 1.73

8

times higher than the corresponding value of the vertical casting specimens.

9

Characteristic example photographs of the fractured surface after loading are

10

shown in Fig. 14. It is clearly seen that the protruded fibers from the surface of the horizontal

11

casting specimen are many more and longer than those of the vertical casting specimen.

12
13
14

BRIDGING LAW CONSIDERING FIBER ORIENTATION
Trilinear model for pullout load versus crack width relationship15

The calculations of the bridging law of the PVA fiber considering the fiber

16

orientation distribution are conducted. The elliptic distribution expressed by the orientation17

intensity and the principal orientation angle is adopted for the PDF estimation of the fiber

18

orientation distribution. The orientation intensity and the principal orientation angle used for

19

calculations are based on these results of the visualization simulations.

20

The pullout properties of the single fiber are required to calculate the bridging

21

law. Several researchers have studied the bond behavior of PVA single fiber to cementitious

22

matrix 20-23 Table 4 lists the results from previously published studies in which the pullout

23

tests of the single fiber were performed. It has been known that the bond behavior of the PVA

24

fiber consists of two stages, i.e., the chemical bond stage and the friction stage 20. The pullout

25

load versus displacement relationships of the PVA fiber commonly exhibit the first peak in12

1

the debonding process of the chemical bond, and slip hardening or softening 21. Table 4 lists

2

the information of the matrix used, the fiber diameter, first peak load, and the maximum load3

in the friction process (second peak). Based on these results, a trilinear model is assumed to4express the relationship between the pullout load and the crack width for a single fiber, as

5

shown in Fig. 15. The pullout load for the first branch, Pa, and it for the maximum, Pmax,

6

corresponds to the first peak load and the second peak load, respectively. As seen in Table 4,

7

there is no test result listed on PVA fibers with a diameter of 0.10 mm (3.910-3 in.), as used

8

in this study. Furthermore, the water to cement ratio used in this study is 0.56, which also9differs from corresponding ratio values in prior studies. Considering the differences of fiber

10

diameters and mix proportions of the matrix, the values of Pa and Pmax are assumed to be 1.5

11

N (0.34 lbf) and 3.0 N (0.67 lbf), respectively. These values correspond to the values of 0.24

12

N (0.054 lbf) and 0.48 N (0.108 lbf) for the same tensile stress of a PVA fiber with a diameter13of 0.04 mm (1.610-3 in.). The values of 0.24 N (0.054 lbf) and 0.48 N (0.108 lbf) are in the

14ranges of the test results reported by Kiyota et al. 22.

15

The crack widths, a and max, are those corresponding to the loads of Pa and Pmax,

16

respectively. These crack widths correspond to the slip-out displacements at the first and17

second peak loads in the pullout test. The slip-out displacements for the two peaks are simply

18

assumed to be 0.1 mm (3.910-3 in.) and 0.3 mm (1210-3 in.) from the test results of Yang et19

al. 23. The crack width becomes twice the slip-out displacement before the maximum load,

20

because the fiber slips out from the both embedded sides. When the pullout load starts to21

decrease at the short embedded side of the fiber, the slip-out displacement at the long22

embedded side decreases 24. To express this phenomenon using a simple trilinear model, the23crack width at the maximum load is assumed to be 1.5 times the slip-out displacement at the24

second peak in the pullout test. As a result, the values of a and max are assumed to be 0.2

25

mm (7.810-3 in.) and 0.45 mm (1810-3 in.), respectively. The softening branch is decided as13

1

the pullout load becomes zero, when the crack width equals the embedded length of the short2

side of the fiber, lb. These assumed values are summarized in Table 5 and illustrated also in

3

Fig. 15.

4

Bridging law simulation method5

The bridging stress can be obtained as the total pullout load of fibers divided by6

the cross-sectional area of the matrix. Moreover, the elliptic distribution is adopted to express

7

the fiber orientation distribution. The snubbing effect 24 and the fiber strength degradation 20

8

are also considered in this study. The snubbing effect exhibits the increment of the pullout

9

load of the fiber due to the edge reaction, when the fiber has the angle with the normal

10

direction of cracking plane. The fiber strength degradation has been adopted for the11

polymeric fibers, which strength decreases when the fiber is pulled out slantingly from its

12

embedded direction.

13

The definitions of the coordinate system and the fiber angle in consideration of14

the snubbing effect and the fiber strength degradation are shown in Fig. 16. The fiber angles,15 and , are the angles between x-axis and the projected lines of the fiber (angle of  to x-

16axis) to x-y and z-x planes, respectively. When the angle  increases, the pullout load also

17

increases owing to the snubbing effect. However, as this angle increases, fiber strength

18

decreases, and the fiber ruptures easily (Fig. 15). The elliptic distribution is considered for

19each of the x-y and z-x planes. Therefore, the formula expressing the bridging stress can be20

given by Eq. (6). Eq. (6) is derived by the summation of the pullout load of the fibers which

21

exist in bridging the crack surface with the probability given in the elliptic distribution. The

22

probabilities for x-y and z-x planes are multiplied to express the probability in 3-D

23

orientation. The pullout load of a single fiber, P, is given by Eq. (7), expressing the snubbing

24

effect and the fiber strength degradation.

14 bridge ( ) 
1


Am
Vf
Af

  Pij ( , )  p xy ( i )  p zx ( j )  p x ( y h , z h )      (y  z )
h

j

(6)

i

P  Ppull  e f   Prup  e  f ' (once exceeded, P  0 )2
3

Pbridge ( )

(7)

where

4

 bridge = bridging stress

5



6

Pbridge = bridging force (= total of pullout load)7Am= cross-sectional area of the matrix

8Vf= fiber volume fraction

9Af= cross-sectional area of a fiber

10

P= pullout load of a single fiber

11

Ppull

= pullout load of a single fiber at a zero fiber angle

12

Prup

= pullout load of a single fiber at rupture at a zero fiber angle

13f= snubbing coefficient

14f'= fiber strength reduction factor

15

p xy , p zx = probability based on elliptic distribution

16

px= probability of fiber distribution along x-axis

17= fiber angle to x-axis18= angle between x-axis and projected line of the fiber to x-y plane19= angle between x-axis and projected line of the fiber to z-x plane= crack width

20

The PDF, px (y, z), gives the probability for the existence of the fiber in the x-axis

21

direction. In this study, px (y, z) is assumed to be constant. This means that the fibers are15

1

randomly distributed along the longitudinal direction of the specimen.

2

The input values for the parameters are listed in Table 5. The orientation3

intensities for the horizontal casting are selected to be 1.5 and 6 for the x-y and the z-x planes,

4

respectively. On the other hand, the corresponding values for the vertical casting are set to 0.5.

5These values are chosen based on the results of the visualization simulation for Vf = 1.5% and6

2.0% (Fig. 8). The principal orientation angles are set to zero for calculation simplification.

7

This value almost agrees with the average value of all the results of the visualization

8

simulation. The calculations were done by using a spreadsheet software.

9Comparison with tension test result

10

The calculated curves showing the variation of the bridging stress (tensile stress)

11

with the crack width are shown in Fig. 17 together with the tension test results, for both the12horizontal and vertical casting specimens. Because the calculated curves exhibit the bridging

13

stress by fibers after cracking, the elastic region before cracking in the tension test (indicated14by dotted line) cannot be compared with the calculated curve. The calculated curves express

15

well the test results after the first peak in the tension test. Based on these calculations, the16

only parameter that differs between the horizontal and vertical casting is the orientation17

intensity. The difference of the fiber orientation intensity identifies a clear influence on the18

bridging law.

19

Fiber effectiveness is also defined to express the effectiveness of the fiber in20

bridging the crack surface. It is calculated as the ratio of the number of fibers crossing the21

crack surface (neither slipping out nor rupturing) to the theoretical number of total fibers in a22

unit volume. The fiber effectiveness is equal to the orientation factor at a crack width of zero.

23

Fig. 17 also shows the calculation results of fiber effectiveness and crack width relationship24both for the horizontal and the vertical casting specimens. The fiber effectiveness values at a25

crack width of zero are 0.544 and 0.315 for horizontal and vertical casting, respectively. The

16

1

difference of the fiber orientation distribution causes this disparity. The fiber effectiveness2

decreases as the crack width increases because of slipping out or because of fiber rupture.

3

The “step” can be seen on the curve, when the fibers rupture more frequently. The balance4

between the increase of the pullout load and the loss of the bridging because of fiber rupture5

leads to the maximum bridging stress. After the end of the “step”, the fiber effectiveness of

6horizontal and vertical casting becomes 0.346 and 0.116, respectively. These values are

7

considered almost equal to the orientation factor after fracture, i.e., the ratio of the fibers that8slipped out from the crack surface. The tension test results shown in Fig. 14 support this9

consideration.

10
11

CONCLUSIONS

12

To investigate the influence of the fiber orientation distribution on the bridging performance13

in PVA-FRCC, visualization simulation using water glass solution and calculation of the14

bridging law considering the fiber orientation distribution were conducted. The main

15

parameter of the investigations is the casting direction of FRCC. The followings are

16

concluded from this study.

17

1. From the visualization simulation, the fibers have a tendency to flow along the18

longitudinal direction in the case of horizontal casting and along the perpendicular

19

direction in the case of vertical casting.

20

2. To evaluate the fiber orientation distribution quantitatively, a new approximation

21

methodology using an elliptic function was introduced. The PDF named elliptic

22

distribution is characterized by the principal orientation angle and the orientation23

intensity.

171

3. From the visualization simulation, while the value of the orientation intensity shows over2

five in the case of horizontal casting, there are the cases that the orientation intensity

3

becomes smaller than 0.5 in the case of vertical casting.

4

4. The bridging stress versus crack width relationship was calculated considering the5elliptic distribution, the snubbing effect and the fiber strength degradation. The

6calculated bridging curves were compared with the results of the tension test in which the7

specimens were fabricated by horizontal and vertical casting. The calculated curves

8

expressed well the test results after first cracking.

9

5. The differences of the fiber orientation distribution clearly indicated an influence on the10

bridging law. Based on the calculation results for the bridging law, it was considered that11

the balance between the increasing pullout load and the loss of the bridging force

12

because of the fiber rupture leads to the maximum bridging stress.

13
14

FUTURE RESEARCH

15

In this study, only one type of mold and matrix was utilized for the visualization

16

simulation. It is considered that the other factors such as casting method, fresh-state

17

properties, flow, vibration, formwork geometry have also influences to the fiber orientation.

18

It is necessary that the influence of these factors on the principal orientation angle and19orientation intensity be clarified. In addition, the influence of the fiber diameter variation to20

the fiber orientation distribution should be investigated. Further experiments of the flow

21

simulation are necessary to study the adaptability of the proposed PDF. If the fiber

22

orientation can be evaluated more quantitatively, the tensile characteristics of FRCC can be

23

estimated more precisely.

24

18

1

ACKNOWLEDGMENTS

2

The authors wish to express their gratitude and sincere appreciation to the Kuraray Co., Ltd.

3for providing the PVA fiber. The tension test was performed in the cooperation with Mr.

4Ryoichi Tsukizaki, a former student in the master’s program of University of Tsukuba. This

5

study was supported by the JSPS KAKENHI Grant Number 24656319.

6
7

REFERENCES

8

1. Bentur, A., and Mindess, S., “Fiber Reinforced Cementitious Composites”, Elsevier

9
10
11

Applied Science, London, UK, 1990.

2. Balaguru, P., and Shah, S. P., “Fiber Reinforced Cement Composites”, McGraw Hill,
New York, 1992.

12

3. Naaman, A. E., and Reinhardt, H. W., Co-Editors, “High Performance Fiber Reinforced

13

Cement Composites: HPFRCC 2”, RILEM Proc., No.31, E. & FN Spon, London, 1996.

14

4. Li, V. C., “From Micromechanics to Structural Engineering – the Design of Cementitious

15

Composites for Civil Engineering Applications”, JSCE Journal of Structural Mechanics

16and Earthquake Engineering, Vol.10, No.2, pp.37-48, 1993.

17

5. Kanda, T., and Li, V. C., “Effect of Fiber Strength and Fiber-Matrix Interface on Crack

18

Bridging in Cement Composites”, ASCE Journal of Engineering Mechanics, Vol.125,

19

No.3, pp.290-299, 1999.

20

6. Li, V. C., and Leung, C. K. Y., “Steady-State and Multiple Cracking of Short Random

21

Fiber Composites”, ASCE Journal of Engineering Mechanics, Vol.118, No.11, pp.2246-

22

2264, 1992.

23

7. Rokugo, K., and Kanda, T., Co-Editors, “Strain Hardening Cement Composites:

24

Structural Design and Performance”, State-of-the-Art Report of the RILEM Technical25

Committee 208-HFC, SC3, Springer, 2013.

19

1

8. Laranjeira, F., Aguado, A., Molins, C., Grünewald S., Walraven, J., and Cavalaro, S.,

2“Framework to predict the orientation of fibers in FRC: A novel philosophy”, Cement and3Concrete Research, 42, pp.752-768, 2012.

4

9. Kanakubo, T., “Tensile Characteristics Evaluation Method for Ductile Fiber-Reinforced

5

Cementitious Composites”, Journal of Advanced Concrete Technology, Vol.4, No.1,

6

pp.3-17, 2006.

7

10. Li, V. C., and Wang, S., “On High Performance Fiber Reinforced Cementitious

8Composites”, JCI Proceedings of the Symposium on Ductile Fiber-Reinforced

9

Cementitious Composites, pp.13-23, 2003.

10

11. Naaman, A., “A Statistical Theory of Strength for Fiber Reinforced Concrete”, Ph.D.

11

dissertation, Massachusetts Institute of Technology, Cambridge, MA, pp.1-196, 1972.

12

12. Stroeven, P., “Stereological Principles of Spatial Modeling Applied to Steel Fiber-

13

Reinforced Concrete in Tension”, ACI Materials Journal, Vol.106, No.3, pp.213-222,

14

2009.

15
16

13. Dupont, D., and Vandewalle, L., “Distribution of steel fibres in rectangular sections”,
Cement & Concrete Composites, 27, pp.391-398, 2005.

17

14. Xia, J., and Mackie, K., “Axisymmetric Fiber Orientation Distribution of Short Straight

18

Fiber in Fiber-Reinforced Concrete”, ACI Materials Journal, Vol.111, No.2, pp.133-141,

19

2014.

20

15. Liu, J., Li, C., Liu, J., Cui, G., and Yang, Z., “Study on 3D Spatial Distribution of Steel

21

Fibers in Fiber Reinforced Cementitious Composites through Micro-CT Technique”,

22

Elsevier Construction and Building Materials, Vol.48, pp.656-661, 2013.

23

16. Torigoe, S., Saito, T., Horikoshi, T., Hamada, T., and Ogawa, A., “Study on Evaluation

24

Method for PVA Fiber Distribution in Engineered Cementitious Composite”,

20

1

Proceedings of the JCI International Workshop on Ductile Fiber Reinforced Cementitious

2Composites, pp.95-101, 2002.

3

17. Kanda, T., Tomoe, S., Nagai, S., Maruta, M., Kanakubo, T., and Shimizu, K., “Full Scale

4

Processing Investigation for ECC Pre-Cast Structural Element”, Journal of Asian

5

Architecture and Building Engineering, Vol.5, No.2, pp.333-340, 2006.

6

18. Japan Society of Civil Engineers, “Standard Specifications for Concrete Structures -2013,

7

Test Methods and Specifications”, JSCE Standard, pp.281-282, 2013 (in Japanese).

8

19. Enomae, T., Han, Y. H., and Isogai, A., “Fiber orientation distribution of paper surface

9

calculated by image analysis”, Proceedings of International Papermaking and

10

Environment Conference, Tianjin, P.R. China (May 12-15), Book 2, pp.355-368, 2004.

11

20. Kanda, T., and Li, V. C., “Interface Property and Apparent Strength of High-Strength12

Hydrophilic Fiber in Cement Matrix”, ASCE Journal of Materials in Civil Engineering,

13

Vol.10, No.1, pp.5-13, 1998.

14

21. Redon, C., Li, V. C., Wu, C., Hoshiro, H., Saito, T., and Ogawa, A., “Measuring and

15

Modifying Interface Properties of PVA Fibers in ECC Matrix”, ASCE Journal of

16

Materials in Civil Engineering, Vol.13, No.6, pp.399-406, 2001.

17

22. Kiyota, M., Mihashi, H., Kanda, T., and Kawamata, A., “Study on Bond Characteristics

18of Fibers in Cementitious Composites”, JCI Proceedings of the Japan Concrete Institute,19

Vol.23, No.2, pp.187-192, 2001 (in Japanese).

20

23. Yang, E. H., Wang, S., Yang, Y., and Li, V. C., “Fiber-Bridging Constitutive Law of

21

Engineered Cementitious Composites”, Journal of Advanced Concrete Technology, Vol.6,

22No.1, pp.181-193, 2008.

23

24. Li, V. C., Wang, Y., and Backer, S., “A Micromechanical Model of Tension-Softening24

and Bridging Toughening of Short Random Fiber Reinforced Brittle Matrix Composites”,

25

J. Mech. Phys. Solids, Vol.39, No.5, pp.607-625, 1991.

26
21

1
2

TABLES AND FIGURES

3

List of Tables:

4

Table 1 Mechanical properties of PVA fiber and target fiber

5

Table 2 Mix proportion of HPFRCC

6

Table 3 Tension test results

7

Table 4 Previous PVA fiber pullout test

8

Table 5 Parameters for bridging law

9

List of Figures:

10

Fig. 1 Flowability test using the funnel (Note: 1 mm = 0.0394 in.)

11

Fig. 2 Mixing of fiber in water glass

12

Fig. 3 Mold for visualization simulation (Note: 1 mm = 0.0394 in.)13

Fig. 4 Camera setup (horizontal casting)

14

Fig. 5 Example of photograph (Vf = 0.1%, horizontal, z-x) (Note: 1 mm = 0.0394 in.)

15

Fig. 6 Image analysis procedures (Note: 1 mm = 0.0394 in.)

16

Fig. 7 Examples of fiber angle histograms

17

Fig. 8 All fiber angle histograms

18

Fig. 9 Approximation method based on elliptic function

19

Fig. 10 Definition of principal orientation angle and orientation intensity

20

Fig. 11 Molds for tensile test specimen

21

Fig. 12 Tensile test specimen (Note: 1 mm = 0.0394 in.)

22

Fig. 13 Tensile stress – crack width curve

23

Fig. 14 Fracture surface after loading (Note: 1 mm = 0.0394 in.)

24

Fig. 15 Trilinear model for pullout load (Note: 1 mm = 0.0394 in.)25

Fig. 16 Definitions of the coordinate system and fiber angle

26

Fig. 17 Calculated bridging law and fiber effectiveness

22

1
2

Table 1 Mechanical properties of PVA fiber and target fiber

Type

PVA
Nylon

Density,g/cm3(lb/ft3)

Length,mm(in.)

Diameter,
mm
(in.)

1.30
(81.2)
1.14(71.2)

12(0.47)
12(0.47)

0.10(3.9  10-3)
0.24(9.3  10-3)

3
4
5

Tensilestrength,
N/mm2
(ksi)
1200
(174)
65
(9.4)

Elastic
modulus,kN/mm2(ksi)
28(4060)-

Table 2 Mix proportion of HPFRCC

Fiber volume
fraction, %Water bybinder ratio

Sand bybinder ratio

2.0

0.39

0.50

Unit weight, kg/m3 (lb/yd3)
Water Cement Fly ash
Sand
380
678
291
484
(641)
(1144)(491)(817)

Cement: High early strength Portland cement
Fly ash: Type II of Japanese Industrial Standard (JIS A 6202)
Sand: Size under 0.2 mm (7.9  10-3 in.)
Super plasticizer: Binder  0.6 %
6
7
8

9
10
11

Table 3 Tension test results

At cracking
Max. after cracking(First peak)(Second peak)
CastingID
Tensile stress, Crack width, Tensile stress, Crack width,
direction
N/mm2
mm
N/mm2
mm
(ksi)
(in.)
(ksi)
(in.)
4.49
0.032
3.70
0.460
TH20-1*1(0.651)(0.0013)(0.537)(0.0181)
4.41
0.034
3.85
0.463
TH20-2*1(0.640)(0.0013)(0.558)(0.0182)
Horizontal
3.17
0.030
2.97
0.446
TH20-3*2(0.460)(0.0012)(0.431)(0.0176)
4.02
0.032
3.51
0.456
Average(0.583)(0.0013)(0.509)(0.0180)
3.53
0.023
1.37
0.328
TV20-1*1(0.512)(0.0009)(0.199)(0.0129)
2.35
0.013
1.55
0.177
TV20-2*2(0.341)(0.0005)(0.225)(0.0070)
Vertical
3.53
0.030
2.09
0.284
TV20-3*2(0.512)(0.0012)(0.303)(0.0112)
3.14
0.022
1.67
0.263
Average(0.455)(0.0009)(0.242)(0.0104)*1: Batch #1, compressive strength 39.2 N/mm2 (5.69 ksi)
*2: Batch #2, compressive strength 41.0 N/mm2 (5.95 ksi)

23

1
2
3

Table 4 Previous PVA fiber pullout test results

Researcher

Water bycementratio

Fiberdiameter,
mm
(in.)

0.27
Kanda et al.
(1998) 20

0.42

0.014
(0.5510-3)

0.62
Redon et al.
(2001) 21

0.30

0.044
(1.710-3)

0.34
Kiyota et al.
(2001) 22

0.42

0.038
(1.510-3)

0.62
Yang et al.
(2008) 23

0.58

0.039
(1.510-3)

4
5

24

First peak
load,
N
(lbf)
0.05 – 0.25
(0.011 – 0.056)
0.12 – 0.20
(0.027 – 0.045)
0.07 – 0.14
(0.016 – 0.031)
0.8 – 1.2
(0.18 – 0.27)
0.3 – 0.6
(0.07 – 0.13)
0.4 – 0.6
(0.09 – 0.13)
0.2 – 0.4
(0.04 – 0.09)
0.3 – 0.6
(0.07 – 0.13)

Second peak
load,
N
(lbf)
1.1 – 1.6
(0.25 – 0.36)
0.5 – 1.3
(0.11 – 0.29)
0.4 – 1.3
(0.09 – 0.29)
0.4 – 0.9
(0.09 – 0.20)
0.5 – 1.0
(0.11 – 0.22)

1
2
3

Table 5 Parameters for bridging law

Parameter
Input value
Remarks*1
First peak load, Pa, N (lbf)
1.5 (0.34)
Crack width at Pa, a, mm (in.)
0.2 (7.8  10-3) 0.1 mm (3.9  10-3 in.)*1  2*1
Maximum load, Pmax, N (lbf)
3.0 (0.67)
Crack width at Pmax, max, mm (in.)
0.45 (18  10-3) 0.3 mm (12  10-3 in.)*1  1.5
2774 (112)
Fiber strength, fu, N/mm (ksi)
1200 N/mm2 (174 ksi)  0.645 *2*2
Snubbing coeffient, f
0.5*2
Fiber strength reduction factor, f’
0.3
Horizontal
1.5
casting
Value near to Vf 1.5 % and 2.0%
Orientation
intensity, kxyvisualizations
Vertical
x-y
0.5
castingplane
Principal orientation0
For calculation simplification *3
angle, r,xy
Horizontal
6
casting
Value near to Vf 1.5 % and 2.0%
Orientation
intensity, kzx
visualizations
Vertical
z-x
0.5
castingplane
Principal orientation0
For calculation simplification *3
angle, r,zx
PVA fiber: 0.10 mm (3.9  10-3 in.) diameter, 12 mm (0.47 in.) length.
*1: Assumed value based on Kiyota et al. (2001) 22 and Yang et al. (2008) 23*2: Assumed value for PVA fiber by Kanda et al. (1999) 5*3: Approximately average value of all Vf visualizations
4
5

25

392
30

422

 70

1
2
3
4
5

6
7
8
9
10
11

Unit: mm 14

Fig. 1 Flowability test using the funnel (Note: 1 mm = 0.0394 in.)(a) PVA fiber(b) Target fiber

Fig. 2 Mixing of fiber in water glass

z
y
x

160

x
y

40

z

12
13
14
15
16

160

40

40

(a) Horizontal casting40

Unit: mm(b) Vertical castingFig. 3 Mold for visualization simulation (Note: 1 mm = 0.0394 in.)26

1
Digital camera

Mold

2
3
4
5
6

Fig. 4 Camera setup (horizontal casting)

40 mm

160 mm

7
8
9
10
11

Fig. 5 Example of photograph (Vf = 0.1%, horizontal, z-x) (Note: 1 mm = 0.0394 in.)

40 mm(a) Cropping

12
13
14
15

(b) Binarization(c) LabellingFig. 6 Image analysis procedures (Note: 1 mm = 0.0394 in.)

27

0.2

0.1

Vf=0.1%
Horizontal
x-y plane

0
-90

2
3
4
5
6

-45
0
45
Fiber angle (degree)

90

0.2

0.1

Vf=0.1%
Horizontal
z-x plane

0
-90

-45
0
45
Fiber angle (degree)90

Fig. 7 Examples of fiber angle histograms

Horizontal castingx-y planez-x plane

Vf
0.2

0.1
%

0.1

0.5
%
Relative frequency

1.0
%

0.1

1.5
%

0.1

45 90

45 90

kxy = 2.7

45 90

r,xy = 2.4 degree

45 90

45 90

k=6.83
r=11.8°

45 90

45 90

k=0.45r=-18.9°

00-90 -45 0 45 90 -90 -45 0 45 90
Fiber angle (degree)kzx = 6.5
kxy = 0.9
r,zx = 15.2 degree
r,xy = -8.5 degree

Fig. 8 All fiber angle histograms

28

45 90

k=0.44
r=29.1°

0-90 -45 0
0.2

0.1

45 90

k=1.76
r=-7.5°

0-90 -45 0
0.2
0.1

45 90

k=2.60r=-24.5°

0-90 -45 0
0.2
0.1

45 90

k=0.55r=22.0°0-90 -45 00.2
0.1

k=0.42r=-18.3°

0-90 -45 0
0.2

0.1

45 90

k=1.00r=-32.6°

0-90 -45 0
0.2
0.1

0.1

k=2.47r=-13.1°

0-90 -45 0
0.2
0.1

45 90

0.2
k=0.34
r=40.5°

0-90 -45 00.2
0.1

k=6.22
r=18.1°

0-90 -45 0
0.2

0.1

45 90

k=11.8
r=7.4°

0-90 -45 0
0.2
0.1

0.1

k=4.01
r=12.4°

0-90 -45 0
0.2
0.1

k=2.03
r=-29.3°

0-90 -45 0

Average

45 90

0.2
k=3.74
r=26.5°

0-90 -45 00.2
0.1

k=0.49r=-41.7°

0-90 -45 0
0.2

2.0
%

45 90

k=1.78r=19.6°

0-90 -45 0
0.2
0.1

0.1

k=4.99r=0.2°

0-90 -45 0
0.2
0.1

Vertical castingx-y planez-x plane0.2
k=4.15
r=4.8°

0-90 -45 0
0.2

7
8
9
10

Relative frequency

Relative frequency

1

45 90

k=0.26
r=9.3°

0-90 -45 0
kzx = 1.1

45 90

r,zx = 5.7 degree

-90

-45
0
45
Fiber angle (degree)

Relative frequency

Relative frequency

1

90

-90

-45
0
45
Fiber angle (degree)

90

a
x*

b
y*
(a) Random orientation

2
3
4
5

x*

r

y*
(b) Longitudinal orientation

Fig. 9 Approximation method based on elliptic function

k >1

k <1

a

a

r
b

x*

r
x*

b
y*
(b) Perpendicular orientationy*(a) Longitudinal orientation

6
7
8
9

Fig. 10 Definition of principal orientation angle and orientation intensity291(a) Horizontal casting

2
3
4
5

(b) Vertical castingFig. 11 Molds for tensile test specimen

R350
110

100

Steel
plate

510
50 130

Universal
joint

130

50

50
100

Pi typedisp.
trans.

Unit: mm

6
7
8
9
10

11
12
13
14

3
2
1
0

0.5
1
1.5
Crack width (mm)

0
2

5

0

Crack width (in.)
0.02
0.04
0.06
TV20
TV20-1 0.6
TV20-2
TV20-3 0.4
Second
peak
0.2

4
3
2
1
0

0

0.5
1
1.5
Crack width (mm)Fig. 13 Tensile stress – crack width curve

30

0
2

Tensile stress (ksi)

Tensile stress (N/mm )TH20
TH20-1 0.6
TH20-2
TH20-3 0.4
Second
peak
0.2

4

0

Crack width (in.)
0.02
0.04
0.06
2

5

0

Tensile stress (ksi)

Tensile stress (N/mm 2)

Fig. 12 Tensile test specimen (Note: 1 mm = 0.0394 in.)

1

Fig. 14 Fracture surface after loading (Note: 1 mm = 0.0394 in.)

Pullout load

2
3
4
5
6

Rupturestrength

=0

Pmax

RupturePa a  maxlb

Crack width

7

Crack width
At Pmax

At Pa
lb

8
9
10
11
12

 :large :medium

Crack width
At slipping out
Fiber

0.1mm

0.3mm

 a =0.2mm

 max =0.45mmlb

MatrixShort
embedded side

Fig. 15 Trilinear model for pullout load (Note: 1 mm = 0.0394 in.)

Crack planeFiber

z-x plane

y x-y plane

13
14
15
16
17

z

x

Fig. 16 Definitions of the coordinate system and fiber angle

31

4
3

0.4
Calc.

2

Test

0

0.5
1
1.5
Crack width (mm)Crack width (in.)
0.02
0.04
0.06

2
0

0
2

TH20
0.6

0.544

0.4
0.346
0.2
0

0.249
0

0.5
1
1.5
Crack width (mm)

3
2

0.2

1
0

Test
0.5
1
1.5
Crack width (mm)Crack width (in.)
0.02
0.04
0.06

0

0

0
2

TV20
0.6
0.4

0.315

0.2

0.116
0

0.5
1
1.5
Crack width (mm)Fig. 17 Calculated bridging law and fiber effectiveness

32

0.6
0.4

Calc.

0

2

TV20

4

0.8
Fiber effectivenessFiber effectiveness

0.8

3
4
5
6

0.2

1
0

0.6

5

Crack width (in.)
0.02
0.04
0.06

0

0.084
2

Tensile stress (ksi)

TH20

Tensile stress (N/mm 2)

5

Crack width (in.)
0.02
0.04
0.06

0

Tensile stress (ksi)

Tensile stress (N/mm 2)1