The 7th International Conference of Asian Concrete Federation“SUSTAINABLE CONCRETE FOR NOW AND THE FUTURE”
30 Oct – 02 Nov, 2016, Hanoi, Vietnam
www.acf2016.vn

EVALUATION OF SIZE EFFECT IN BENDING CHARACTERISTICS OF DFRCC
BASED ON BRIDGING LAW
Y. Ozu1, K. Watanabe1, A. Yasojima2, and T. Kanakubo3
12

Master Program, Dept. of Engineering Mechanics and Energy, University of Tsukuba, Tsukuba, Japan
Associate Professor, Dept. of Engineering Mechanics and Energy, University of Tsukuba, Tsukuba, Japan
3Professor, Dept. of Engineering Mechanics and Energy, University of Tsukuba, Tsukuba, Japan

ABSTRACT: The purpose of this study is to evaluate the size effect in bending characteristics of ductile
fiber-reinforced cementitious composites (DFRCC) with PVA fibers using the bridging law consideringfiber orientation. Four-point bending test with experimental factor of specimen size was conducted.
From the test results, the size effect that bending strength decreases with increasing of the specimen
size was recognized. The bridging law considering fiber orientation was calculated based on the previous
study, and the section analysis using the bridging law was performed. From the analysis results, the size
effect can be evaluated by the bridging law which shows the tendency that the axial orientation of thefibers becomes stronger with decreasing of specimen size.

1INTRODUCTION

Ductile fiber-reinforced cementitious composites (DFRCC), in which short fibers with a certain
percentage of volume fraction are mixing in mortar or concrete, show the improved tensile and bending
characteristics of cementitious composites. DFRCC is applied to the structural elements expecting highperformance in ductility and tensile characteristic, because short fibers bridge the crack after first
cracking and transfer the tensile force. Some previous studies indicated that the tensile characteristicsof DFRCC is strongly affected by fiber orientation and distribution in matrix (e.g. Laranjeira et al. (2012)).
However, there is little study evaluating the effect of fiber orientation to tensile characteristics of DFRCC
quantitatively.
The tensile performance of DFRCC is characterized by the bridging law (tensile stress – crack width
relationship). The bridging law can be expressed by the sum of the pullout properties of single fiber. The
mechanical characteristics of DFRCC can be evaluated accurately using the bridging law considering fiber
orientation. On the other hand, structural elements using DFRCC show the size effect that the strength
decreases with increasing of cross-sectional size (Kanakubo et al. (2007)). Moreover, it is considered thatthe size effect is also caused by two-dimensional fiber orientation. The strength and the ductility
increase for small size specimens due to two-dimensional fiber orientation along the wall of the mold.
In this study, the four-point bending tests for different sizes of specimens using DFRCC with PVA fibers
are conducted. The size effect observed in the experiment is evaluated quantitatively using the bridging

law considering fiber orientation.
2CALCULATION OF BRIDGING LAW CONSIDERING FIBER ORIENTATION

Bridging law can be expressed by the sum of the pullout behavior of each single fiber bridging crack
plane considering bond properties between single fiber and matrix, including the snubbing effect andthe fiber strength degradation. The snubbing effect shows the increment of the pullout load due to theedge reaction when fiber is embedded to matrix with inclined angle. The fiber strength degradation
expresses the decrement of fiber strength because the surface of fiber is roughed by the asperity of thematrix when fiber has inclined angle. The fiber orientation in the crack plane is evaluated by probability
density function (PDF) which expresses the distribution of orientation angle of the fibers that bridge thecrack (Kanakubo et al. (2016)). Figure 1 shows the definition of coordinate system and orientation angleof fiber.
2.1

Pullout property of single fiberFigure 1. Definition oforientation angle.

Figure 2. Pullout model of singlefiber.

Figure 3. Definition ofembedded length.

Table 1. Parameter for the pullout model of single fiber
ParameterFirst peak load Pa (N)Crack width at Pa (mm)Maximum load Pmax (N)Crack width at Pmax (mm)Snubbing coefficient f
Fiber strength σfu (MPa)Fiber strength reduction factor f‘
Fiber diameter df (mm)Fiber length lf (mm)Used value
1.5
0.2
3.0
0.45
0.5
569
0.3
0.10
12The
pullout tests of single fiber have been conducted by several researchers to evaluate the bond propertiesof fiber. Figure 2 shows the model of the relationship between the pullout load and the pullout
displacement of single fiber used in current study. Table1 shows the value of each parameter for thepullout model of single fiber. The relationship is modeled by tri-linear model considering chemical bondand friction between PVA fiber and matrix (Redon et al. (2001)). The first peak load Pa corresponds tothe debonding of the chemical bond and pullout load increases to the maximum load Pmax. After Pmax,single fiber is pulled out completely when the pullout displacement becomes equal to the embedded
length lb. Figure 3 shows the definition of the embedded length lb that is calculated by orientation angle
ψ in Figure 1 and the distance between the midpoint of fiber and assumed crack surface. The first peakload, the maximum load, and each pullout displacement are determined based on the previous study(Kanakubo et al. (2016)). The values of snubbing coefficient, fiber strength, and fiber strength reduction
factor are assumed to be 0.5, 569MPa, 0.3, respectively (Kanda and Li (1999), Asano (2014)).

2.2

Fiber orientation distribution

The elliptic function has been introduced as the PDF in order to evaluate the fiber orientation
distribution quantitively in the previous study (Kanakubo et al. (2016)). Figure 4 shows the example ofrelative frequency distribution by the visualization simulation of PVA fibers in the matrix using a sodium
silicate solution (water glass) which is clear and colorless, and has high viscosity such as the matrix ofDFRCC (Miyaguchi et al. (2014)). The solid line in Figure 4 shows the elliptic distribution. The ellipticdistribution is expressed by the orientation intensity k and the principal orientation angle θr. The relative
frequency for each orientation angle of fiber orientation distribution is transformed into plane
coordinates (x*-y*) with the argument to be equal to orientation angle. The relative frequency in x*-y*
coordinates are approximated by the ellipse function using the least square method. The ratio of theradii of the approximated ellipse (a/b) is defined as orientation intensity, k, and the angle between theradius a and the axis x* is defined as principal angle, θr. The range of the value of the principal angle θr is-45° to +45°. The tendency of fiber orientation toward the direction of the principal angle is expressedby the value of k. When the fibers tend to orient to θr, the value of k becomes larger than 1, and when
they orient to the perpendicular direction to θr, the value of k becomes smaller than 1. The randomorientation is shown as the value of k = 1.

Figure 4. Elliptic distribution for fiber orientation.

2.3

Calculation of bridging law

The bridging law considering fiber orientation is calculated by combining the pullout properties of thesingle fiber with the elliptic distribution as the PDF which evaluate the fiber orientation quantitatively as
given by equation (1).
Pbridge   Pij ( , )  p xy ( i )  p zx ( j )  p x ( yh , z h )      (y  z )
hj(1)

i

Pbridge= bridging force

Pij(δ,ψ)= tensile force of single fiber

pxy(θi), pzx(φj)= probability based on elliptic distribution for x-y plane and z-x planepx(yh,zh)= probability of fiber distribution along x axis (constant)θ, φ= refer to Figure 1Figure 5 presents the example of bridging law calculated considering the fiber orientation. The calculated
bridging force is converted to tensile stress dividing the force by cross-sectional area. The parameters forcalculation are already shown in Table1, and principal angle θr is assumed to be equal to 0° (axialdirection) in the calculation. The curves in the figure show the bridging laws for each value of orientation
intensity k (k=0.4, 1.0, 4.0). Figure 5 shows the stress only by the bridging law, that means the crackstrength of the matrix is not expressed in the figure. As shown in Figure 5, tensile stress remarkably
decreases after the maximum load due to the rupture of PVA fibers. After that the degradation of theslope changes gentle. The calculation result for larger k, that expresses the longitudinal orientation offibers, gives larger strength. That means that the fiber orientation greatly affects the bridging law andthe tensile behavior.

Tensile stress (MPa)The calculated bridging law is modeled by the tri-linear model (Figure 6) in order to apply the bridging
law to the section analysis explained after. The model is defined by two characteristic points, namely, thetensile strength, σmax , and the changing point of degradation, σ2. The corresponding crack width for eachpoint is defined as δmax, and δ2. The tensile stress becomes to be equal to zero when the crack widthreaches the half length of fiber (completely pulled out).
3k=4
k=1
k=0.4

2
1001
Crack width (mm)2Figure 5. Calculation results of bridging law.

3OUTLINE OF BENDING TEST

3.1

Materials

Figure 6. Modeling of bridging law.

Table 2 shows mechanical property of PVA fiber and mix proportion of DFRCC applied in this study. The
volume fraction of PVA fiber is 2.0%. A total of seven mixing batch was conducted by the same mix
proportion.
3.2Specimens

The dimensions of specimens are shown in Figure 7. The main experimental parameter is the crosssectional size. Specimens have square or rectangular cross-section. Three series of specimens, (a) smallspecimen (40x40x160mm), (b) middle specimen (100x100x400mm), and (c) large specimen(180x280x1680mm) were tested. Three small and middle specimens are manufactured by each mixing
batch. The number of large specimen is one. Table 3 shows the compression test results of cylinder(φ100-200mm) test pieces and fresh properties.
Table 2. Mechanical properties of PVA fiber
Length of fiber
Diameter of fiber(mm)(mm)
120.10Tensile strength(MPa)Elastic modulus(GPa)1200

28Table 3. Fresh properties and compression test results
TemperatureAir content
Compressive strength(%)
(MPa)
(°C)
24.5

1.8

45.6

Elastic modulus
(GPa)
13.9

4040402040

Castingplane

PI typedisp. trans.

25402050x(a) Small specimen.

100

100

10050

100

Castingplane

7010050

PI typedisp. trans.

x(b) Middle specimen.

280x550

PI typedisp. trans.

50

180
Castingplane

280

45 190 45550

50(c) Large specimen.
Figure 7. Dimensions of specimens.

3.3

Method of loading and measurements

Four-point bending tests were carried out using a universal testing machine of 500kN capacity for smalland middle specimens, and using a universal testing machine of 2000kN capacity for the large specimen.
The measurement items for all specimens were load and axial deformation to calculate averagecurvature of pure bending region. In addition, six pi type displacement transducers of gauge length of
50mm were set along the upper and lower position in the case of the large specimen.
4TEST RESULTS

The examples of the specimens after loading are shown in Figure 8. The loading points are indicated byarrows. In the small specimens, there was only one crack within pure bending region, and the crack wasgradually opened, after that, specimens fractured. On the other hand, in the middle specimens and thelarge specimen, there were one or two cracks within pure bending region, after that, either one of thetwo cracks was opened and load decreased.
In order to compare the effect of the difference of the specimen size, bending stress, which is bending
moment divided by section modulus, and bending strain, which is curvature multiplied by depth of thespecimen, are defined. Figure 9 shows the relationships between bending stress and bending strain foreach specimen size except the specimens that fractured at the out of pure bending region. As shown inFigure 9, in the case of small specimens, after decreasing the load due to cracking, the load graduallyincreased again. After the peak load, the load moderately decreased. In the case of middle and large
specimens, the load degradation at cracking was relatively small, the load slightly decreased after peakload. The higher maximum load and the deformability were observed for the specimens in which thetwo or more cracks took place. The size effect that bending strength decreases with increment ofspecimen size is recognized.

280mm

100mm

(a) Small specimen.

(b) Middle specimen.

(c) Large specimen.

6420Batch No.1
Batch No.2
Batch No.3

0Batch No.4
Batch No.5
Batch No.6

2468
Bending strain(%)(a) Small specimen.

8Batch No.1
Batch No.2
Batch No.3
Batch No.4

6Batch No.5
Batch No.6
Batch No.7

4200

Bending stress(MPa)8Bending stress (MPa)

Bending stress(MPa)Figure 8. Specimens after loading.

2468
Bending strain (%)(b) Middle specimen.

8Batch No.1

642002468
Bending strain(%)(c) Large specimen.

Figure 9. Bending stress - bending strain curve.

5SECTION ANALYSIS

5.1

Stress – strain model of DFRCC

The section analysis employing the bridging law of DFRCC considering fiber orientation is conducted foreach size of specimens, and the analysis results are compared with test results. Figure 10 and Table 4
show the stress-strain model of DFRCC and the values of the parameters of the model, respectively. The
compressive stress-strain model is the parabolic model and the compression test results of the testpieces (Table 3) are utilized as the parameters of the model. The tensile stress-strain model is derived
from the model of the bridging law (Figure 6).

Figure 10. Modeling of stress - strain curve of DFRCC.
Table 4. Parameters of the stress - strain modelSpecimen size
Orientation intensity kCompression strength σc (MPa)

Small
0.4
1.0
43.6

4.0

Middle
0.4
1.0
43.6

4.0

Large0.4
1.0
43.6

4.0Strain at σc, εc (%)
Average crack spacing (mm)Maximum tensile stress σmax (MPa)Strain at σmax, εmax (%)
Second peak tensile stress σ2 (MPa)Strain at σ2, ε2 (%)
Tensile ultimate strain εtu (%)

0.45
40
1.54
0.45
0.27
1.13
15.0

2.11
0.49
0.62

3.13
0.64
1.85

0.45
65.4
1.54
0.28
0.27
0.69
9.18

2.11
0.30
0.62

3.13
0.39
1.85

0.45
140
1.54
0.13
0.27
0.32
4.29

2.11
0.14
0.62

3.13
0.18
1.85

Analysis results

5.2.1Analysis for random orientation

Bending strength (MPa)5.2

86Test result
k=142040 100
280
Specimen depth (mm)Figure 11. Analysis results (k=1).

Bending strength(MPa)The parameters of the tension model based on the bridging law are determined by orientation intensity
k. Moreover, the crack width of the bridging law is transformed into the strain dividing by the averagevalue of crack spacing observed in the bending test. The crack spacing is obtained as the length of purebending region divided by the number of cracks. In this paper, θr is assumed to be equal to 0° (axialdirection of specimens).

864
20

Test result
k=0.4
k=1.0
k=4.040 100
280Specimen depth(mm)Figure 12. Analysis results (k=4, 1, 0.4).

Figure 11
shows the comparison of bending strength between test results and analysis results in the case ofrandom orientation (k=1). The bending test results are plotted by square marks with the error bars thatexhibit the standard deviations of test results for specimens of each size. The analytical bending strength
decreases with increasing specimen size in spite of using the same bridging law (same value of k),
because the strain gradient in cross-section increases with the increment of the cross-sectional size. As
shown in the figure, the test result mostly corresponds to the analysis result only for the middle
specimen.
5.2.2

Analysis considering fiber orientation

Orientation intensity k for small specimens is generally 4 in the visualization simulation in the previous
study (Miyaguchi et al. (2014)). Therefore, the section analysis is carried out again using the different
value of k. The assumed value for k is set to 4 for the small specimen, and set to 0.4 for the large
specimen. The k value of 4 and 0.4 shows that the fibers tend to orient longitudinally and toward
perpendicular to axial direction of a specimen, respectively. The comparison between test results andanalysis results is shown in Figure 12. The test results of small specimen, middle specimen, and large
specimen show good agreements with analysis results. These results indicate that PVA fibers in smallspecimens tend to orient along axial direction, and there is a tendency that orient toward perpendicular
to axial direction with increasing of specimen size.

From the observation of the flow of the DFRCC pouring in the case of large specimen, the matrix spread
concentrically from the position of inflow of DFRCC. This observation suggested that fibers orient
concentrically and tend to turn toward perpendicular to axial direction.
6CONCLUSION

In this study, the size effect in bending characteristics of DFRCC was confirmed experimentally. The
bridging law considering fiber orientation is calculated based on the previous study and the tri-linear
model of the bridging law is proposed. The size effect in the bending strength can be evaluated by thesection analysis using the tensile stress-strain model of DFRCC that shows the tendency of the stronger
orientation toward the axial direction with decreasing specimen size.
7

ACKNOWLEDGEMENT

This study was supported by the JSPS KAKENHI Grant Number 26289188.
8REFERENCES

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Cementitious Composites”, Doctoral thesis of University of Tsukuba. (in Japanese)
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under Extreme Loading, Proceedings of Protect 2007.
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Redon, C., Li, V.C., Wu, C., Hoshiro, H., Saito, T., and Ogawa,A. (2001) “Measuring and Modifying InterfaceProperties of PVA Fibers in ECC Matrix”, ASCE Journal of Materials in Civil Engineering, Vol.13, No.6: 399-406.