Influence of Matrix Strength on Tensile Performanceof DFRCC with Bundled Aramid Fiber（集束型アラミド繊維を用いた DFRCC の引張性能に及ぼすマトリクス強度の影響）

SHI HAOHUI(Master’s Program in Engineering Mechanics and Energy)

Advised by Toshiyuki Kanakubo
Submitted to the Graduate Schoolof Science and Technology
Degree Programs in Systems andInformation Engineeringin Partial Fulfillment of the Requirements
for the Degree of Master of Engineering
at the University of Tsukuba

March 2022

Abstract
Fiber-Reinforced Cementitious Composite (FRCC) is cementitious material reinforced with shortdiscrete fibers to improve brittle behavior of composite, especially in tensile and bending field. FRCC
showing a deflection hardening behavior under bending condition is defined as Ductile Fiber-Reinforced
Cementitious Composite (DFRCC). The main advantage of DFRCC is the high tensile performancebrought by the bridging effect of fibers across cracks. In order to utilize its advantage in actual structuralelements, it is essential to investigate tensile performance of DFRCC according to various factors anddevelop the evaluation methodologies for it.
This study aims to investigate the influence of matrix strength on the tensile performance of DFRCC
with bundled aramid fiber. A total of 45 specimens varying matrix strength and fiber volume fraction
were subjected to uniaxial tension test. Most of the DFRCC specimens in tensile failure showed multiple
cracks after loading. Number of cracks increased with increasing of matrix strength. Furthermore, theload at first cracking and the maximum load increased with increasing of matrix strength.
By using the pullout characteristic of bundled aramid fiber obtained from previous study, theoreticalbridging law was calculated to adapt with the experimental results. According to the calculation, themodified individual fiber pullout model, in which the maximum pullout force is given by the
proportional relations with the compressive strength of matrix, could express the uniaxial tension test
results well.
In addition, based on the experimental and calculation results, bridging law model, in which featurepoints were expressed by the function of fiber orientation intensity and matrix strength was proposed.
The modeled bridging law was applied to the section analysis for 4-point bending test to verify its
adaptability. According to the comparison, the maximum bending moment of experimental results isfrom 0.910 to 1.105 times of that of analysis results.

Contents
Chapter 1 Introduction............................................................................................................................. 1
1.1 Research Background ................................................................................................................ 1
1.1.1 Ductile Fiber-Reinforced Cementitious Composite (DFRCC) ....................................... 1
1.1.2 Aramid fibers applied in DFRCC .................................................................................... 3
1.1.3 Evaluation for Tensile Performance of DFRCC (Fiber Bridging Law) .......................... 4
1.2 Research Objective .................................................................................................................... 5
Chapter 2 Uniaxial Tension Test ............................................................................................................. 6
2.1 Introduction................................................................................................................................ 6
2.2 Experiment Outline .................................................................................................................... 7
2.2.1 Specimens ....................................................................................................................... 7
2.2.2 Applied Materials ............................................................................................................ 8
2.2.3 Loading and Measurement ............................................................................................ 10
2.3 Experiment Results .................................................................................................................. 11
2.3.1 Compressive Properties ................................................................................................. 11
2.3.2 Failure Mode under Tension ......................................................................................... 122.3.3 Crack Pattern and Crack Width ..................................................................................... 13
2.3.4 Load P - Crack width w Relationship ........................................................................... 16
2.3.5 Fibers on Fracture Surface ............................................................................................ 21
Chapter 3 Calculation of Bridging Law and Modeling ......................................................................... 23
3.1 Introduction.............................................................................................................................. 23
3.2 Calculation of Bridging Law ................................................................................................... 24
3.2.1 Calculation Method ....................................................................................................... 24
3.2.2 Calculation Parameters.................................................................................................. 25
3.2.3 Calculation Results........................................................................................................ 26
3.3 Modeling of Bridging Law ...................................................................................................... 31
3.4 Adaptability of Modeled Bridging Law ................................................................................... 46
3.4.1 4-point Bending Test ..................................................................................................... 46
3.4.2 Section Analysis ............................................................................................................ 48
3.4.3 Comparison of Analysis and Experimental Results ...................................................... 50
Chapter 4 Conclusion ............................................................................................................................ 52
Acknowledgements ............................................................................................................................... 53
Reference ............................................................................................................................................... 54

List of Figures
Figure 1.1 DFRCC under bending condition ........................................................................................2
Figure 1.2 Bundling of original yarns of aramid fiber ...........................................................................3
Figure 2.1 Dimension of specimen for uniaxial tension test ..................................................................7
Figure 2.2 Visual appearance of bundled aramid fiber ..........................................................................8
Figure 2.3 Setup of loading and measurement for uniaxial tension test ...............................................10
Figure 2.4 Failure modes of specimen under tension ..........................................................................12
Figure 2.5 Examples of crack pattern (tensile failure) .........................................................................14
Figure 2.6 Load P – total crack width w relationship ..........................................................................17
Figure 2.7 Relationship between load at first crack P1 and matrix strength fc ......................................20
Figure 2.8 Relationship between maximum load Pmax and matrix strength fc ......................................20
Figure 2.9 Examples of the visual appearance of fracture surface (Fc36) ...........................................21
Figure 3.1 Tensile load per fiber Pind. - crack width w relationship ...............................................27-29
Figure 3.2 Relationship between coefficient α and matrix strength fc .................................................30
Figure 3.3 Calculation results of bridging law (fc = 45MPa) ...............................................................31
Figure 3.4 Bilinear model for bridging law .........................................................................................32
Figure 3.5 Modeled bridging law (fc = 45MPa) ...................................................................................32
Figure 3.6 Relationship between each parameter and orientation intensity k (fc = 45MPa) .................33
Figure 3.7 Calculated and modeled bridging law ..........................................................................34-42
Figure 3.8 Relationship between each parameter and orientation intensity k ...............................34-42
Figure 3.9 Calculation results of bridging law (fc ≤ 30MPa) ...............................................................43
Figure 3.10 Relationship between each coefficient and matrix strength fc ..........................................45
Figure 3.11 Image of 4-point bending test ...........................................................................................47
Figure 3.12 Stress σ - strain ε model applied in section analysis ..........................................................49
Figure 3.13 Bending moment M - curvature φ relationship .................................................................51

List of Tables
Table 2.1 List of specimens ...................................................................................................................7
Table 2.2 Dimension and mechanical properties of fiber ......................................................................8
Table 2.3 Mix proportions of cementitious matrix ................................................................................9
Table 2.4 Compressive properties (Uniaxial tension test) ...................................................................11
Table 2.5 Number of cracks in testing area .........................................................................................15
Table 2.6 Load at first crack P1 and crack width at first crack w1 .........................................................18
Table 2.7 Maximum load Pmax and total crack width at maximum load wmax .......................................19
Table 2.8 Number of fibers on fracture surface ...................................................................................22
Table 3.1 Parameters for the calculation of bridging law ....................................................................25
Table 3.2 Maximum load per an individual fiber Pind.max ...............................................................27-29
Table 3.3 Coefficients of parameters for bilinear model .....................................................................43
Table 3.4 Compressive properties (4-point bending test) ....................................................................47
Table 3.5 Parameters for stress σ - strain ε model applied in section analysis ......................................49
Table 3.6 Maximum bending moment Mmax of experimental and analysis results ...............................50

Chapter 1 Introduction
1.1 Research Background1.1.1 Ductile Fiber-Reinforced Cementitious Composite (DFRCC)
Fiber-Reinforced Cementitious Composite (FRCC) is cementitious material in which short discrete
fibers with a certain volume fraction are mixed in mortar matrix to improve brittle behavior of composite
especially in tensile and bending field. Compared with traditional mortar matrix, FRCC is expected tohave high performance in ductility because of the fiber bridging through cracks which transfers thetensile force as shown in Figure 1.1(a). In addition, FRCC is expected to bring high durability when it
is used in Reinforced Concrete (RC) structures because fibers can control the crack openings in matrix
which prohibit the penetrations of aggressive attacks to deteriorate the internal reinforced rebars and
FRCC itself.
In the past decades, various types of FRCC have been introduced and studied by a number of
researchers. As a type in FRCC categories with further improved performance, Ductile Fiber-Reinforced
Cementitious Composite (DFRCC) is developed. As a significant characteristic, DFRCC shows adeflection hardening behavior under bending condition which means stress increases after the initial
cracks generate [1]. In addition, DFRCC shows multiple cracking behavior which is shown in Figure
1.1(b). The high ductility and durability of DFRCC is achieved by the bridging effect of individual fibers
in the mortar matrix.
By applying the high ductility and durability of DFRCC to the structural elements, the reduction of
the damage and crack width reduction effect have been reported by a lot of research outcomes (e.g. [2]).
Therefore, the application of DFRCC has the potential to bring higher performance (such as: weightsaving, high durability) for conventional RC structures and realize structural design for complicatedstructures.
It has been reported that High Performance Fiber-Reinforced Cement Composite (HPFRCC), whichis included in the category of DFRCC and showing strain-hardening behavior under uniaxial tension,
has been applied in high-rise housing in crucial parts such as coupling beams [3]. However, there arefew examples of applying DFRCC for structural elements expecting its fiber bridging effect. The current
condition is that DFRCC is mainly expected for controlling effect of crack openings and prevention
effect of peelings, and as a result, it is mainly used for non-structural elements [4].
As one of the main reasons why DFRCC is hardly used for structural elements, a normative evaluationof its tensile performance has not been fully developed. It is extremely difficult to evaluate the tensile
performance of DFRCC quantitatively because bond performance of individual fibers with matrix andorientation of fibers in matrix would change due to various factors.

1

(a)
(b)
Figure 1.1 DFRCC under bending condition:(a) Fiber bridging through crack; (b) Multiple cracking behavior21.1.2 Aramid fibers applied in DFRCC
According to various studies, steel fibers or polymeric fibers such as polyethylene (PE), polyvinyl
alcohol (PVA), and polypropylene (PP) fibers have been utilized in FRCC. While FRCC mixing withsteel fibers commonly shows tension softening behavior after initial cracking, polymeric fibers arecommonly used rather than steel fibers in DFRCC which shows higher ductility.
Aramid fiber is known as one of the polymeric fibers that have high tensile strength, durability, andheat and chemical resistance. It has been reported that aramid fiber has been used for the strengthening
for RC structures by the external bonding of a fiber sheet. However, few research studies can be foundconcerning FRCC mixed with discrete aramid fibers [5-6]. Since a commercially provided single aramid
fiber has a small diameter of 12 μm, it cannot be expected that the aramid fibers and the cementitious
matrix have a strong bond strength [7]. In the case of PVA fiber, it has been considered that the alcoholgroup in a PVA molecule leads to the good bond performance with the cementitious matrix. However,as for other types of polymeric fibers, the smooth surface of an individual fiber cannot generate a strong
bond resistance.
For these reasons, this study has been focusing on bundled aramid fiber, which is made with abundling of original yarns of aramid fiber as shown in Figure 1.2. Bundled aramid fiber shows a roughsurface, and it is expected to have good bond performance with the cementitious matrix by mechanicalresistance.

Figure 1.2 Bundling of original yarns of aramid fiber3

1.1.3 Evaluation for Tensile Performance of DFRCC (Fiber Bridging Law)
The high tensile performance of FRCC is brought by the bridging effect of fibers across cracks after
initial cracking of the matrix. Therefore, fiber bridging stress - crack width relationship called bridging
law has been studied by lots of researchers to evaluate tensile performance of FRCC. According toprevious studies, bridging law can be obtained experimentally (e.g., from a uniaxial tension test). On
the other hand, theoretical solution of bridging law based on micromechanics has been introduced bysome researchers.
Since the bridging effect of fibers is strongly influenced by the fiber types and dimensions, the pullout
behavior of an individual fiber from cementitious matrix has been investigated, to reflect these factors
on bridging law. A number of researchers have conducted pullout tests for various types and dimensionsof fibers, e.g., steel fiber [9], bundled aramid fiber [10], nylon and polypropylene (PP) fibers [11].
Bridging law can be obtained by an integral calculus of forces carried by individual bridging fibers,considering the orientation and distribution of fibers in matrix [8]. In order to evaluate the orientationand distribution of fibers in matrix quantitatively, Kanakubo et al. has introduced a probability density
function (PDF) for fiber inclination angle and fiber centroidal location expressing by an elliptic function[12].
Bridging laws for PVA-FRCC [13] and bundled aramid-FRCC [10] have also introduced by
Kanakubo et al. From the results, both calculated bridging laws showed good agreements with the resultsof uniaxial tension test. In addition, the calculated bridging laws have been simply modeled by several
researchers, e.g., tri-linear model for PVA-FRCC [14], bi-linear model for bundled aramid-FRCC [15].
These models make it possible to evaluate the bridging effect in various types of FRCC members
approximately.

4

1.2 Research Objective
In recent years, requirements for concrete structures have diversified and advanced. Consequently,
the performance required for RC structures is also increasing. In addition, it is considered that theimportance of extending the life cycle of structures, which improves global environment problems,
would be further enhanced in the future. For such social requirements, DFRCC would be useful and has
the potential to solve various tasks when applying to structural elements. Therefore, it is essential toinvestigate the mechanical performance especially tensile performance of DFRCC according to various
factors and develop the evaluation methodologies for its mechanical performance.
In the past decades, a number of researchers have investigated the tensile performance of FRCC via
several factors. Li et al. [16] and Park et al. [17] has introduced the tensile performance of Ultra-HighPerformance Concrete (UHPC) with different hybrid fibers. Kanakubo et al. has investigated theinfluence of fiber orientation on bridging performance of PVA-FRCC [12]. However, it can be seen thatthe research studies on mechanical performance of FRCC categories mainly focused on the influence offibers (e.g., types, adding amount, orientation), while few research studies could be found considering
the influence of mortar matrix.
According to this situation, this study focuses on the property of mortar matrix applied in DFRCC.
The main objective of this study is to investigate the influence of matrix strength on the tensile
performance of DFRCC with bundled aramid fiber. In this study, matrix strength and fiber volume
fraction are considered as main parameters. Uniaxial tension test was conducted for DFRCC rectangularprism specimens with slits to find out the relationship between tensile performance and the twoparameters.
In addition, by using the pullout characteristic of bundled aramid fiber obtained from previous study[10], theoretical bridging law was calculated to adapt with the experimental results. The calculated
bridging law was simplified to a bilinear model, in which feature points are expressed by the functionof fiber orientation intensity and matrix strength. After that, the modeled bridging law was applied tothe section analysis under bending condition. Comparison between analysis results and experimental
results obtained from 4-point bending test was discussed to verify the adaptability of modeled bridging
law.

5

Chapter 2 Uniaxial Tension Test
2.1 Introduction
In this chapter, the uniaxial tension test is conducted for DFRCC rectangular prism specimens withslits using bundled aramid fibers to investigate the influence of matrix strength on tensile performance.
The tensile performance is compared to each other through the load P – total crack width w relationships
obtained from the experiment. Furthermore, the crack pattern and fibers on the fracture surface areobserved and discussed.

62.2 Experiment Outline
2.2.1 Specimens
List of specimens are shown in Table 2.1. Matrix strength and fiber volume fraction are considered
as main experimental parameters in this study. The target compressive strength of the mortar matrix was
set to 24MPa, 36MPa, and 48MPa for Fc24, Fc36, and Fc48 series of specimens, respectively. Fibervolume fraction was varied as none (mortar), 1%, 2% for each matrix strength. Therefore, 9 series of
specimens were determined for uniaxial tension test.
Figure 2.1 shows the dimension of specimen for uniaxial tension test. Specimen is DFRCC
rectangular prism with two bolts (M20) embedded at both ends to transfer the tensile load. In order tocontrol the position of crack opening, two slits were set on both narrow sides in the middle of specimen.
For each series, five specimens were manufactured and tested.

SeriesFc24-N
Fc24-1%
Fc24-2%
Fc36-N
Fc36-1%
Fc36-2%
Fc48-N
Fc48-1%
Fc48-2%

No.
1-5
1-5
1-5
1-5
1-5
1-5
1-5
1-5
1-5

Table 2.1 List of specimens
Target matrix strength [MPa]
24

36

48

Fiber volume fraction [%]
None
1
2
None
1
2
None
1
2

Figure 2.1 Dimension of specimen for uniaxial tension test

7

2.2.2 Applied Materials
The aramid fiber applied in this study is a bundled fiber. The original yarns of aramid fibers with anominal diameter of 12μm are twisted to form a thick individual fiber and sized not to unravel in FRCC.
Table 2.2 shows the dimension and mechanical properties of fiber applied in this study and Figure 2.2
shows its visual appearance. The unit weight of bundled aramid fibers mixed in DFRCC for volume
fraction of 1% and 2% is 13.9kg/m3 and 27.8kg/m3, respectively.
Table 2.3 shows the three mix proportions of cementitious matrix applied in this study. Materials in
the three mix proportions are different in the ratio of cement to fly ash to obtain different target compressive
strength. Since the fresh DFRCC shows self-consolidating property, fresh DFRCC was poured from one endof the mold and allowed to flow naturally until the mold was full filled. The specimens were cured in thenatural environment.
Table 2.2 Dimension and mechanical properties of fiber
LengthDiameter
Tensile strength
Elastic modulus
Fiber type
[mm]
[mm]
[MPa]
[GPa]
Bundled aramid fiber
30
0.50
3432*
73*
* Properties obtained from original yarns(a)(b)
Figure 2.2 Visual appearance of bundled aramid fiber:(a) Length; (b) Diameter8Series

W/C

Fc24
Fc36
Fc48

0.785
0.560
0.436

Table 2.3 Mix proportions of cementitious matrix
Unit weight [kg/m3]
W/B
FA/B
W
C
FA
S
SP
0.392
0.500
380
484
484
484
6
0.392
0.300
380
678
291
484
6
0.392
0.100
380
872
97
484
6

where,
B = binder (= C + FA),W = water,
C = cement (High early strength Portland cement),
FA = fly ash (Type II of Japanese Industrial Standard (JIS A 6202)),S = sand (size under 0.2 mm),
SP = high-range water-reducing agent (= B × 0.6%),
TH = thickener,
SR = shrinkage reducing agent (= C × 2%).

9

TH
2
2
2

SR
9.68
13.56
17.44

2.2.3 Loading and Measurement
Uniaxial tension test was conducted using a universal testing machine with the capacity of 2MN.
Figure2.3 shows the setup of the loading and measurement for uniaxial tension test. Since the increasing
external moment caused by setup irregularity and local fracture caused by secondary moment would be
an inevitable factor to the experiment, pin-fix ends were applied at the boundaries to minimize possible
effects to the results. Two displacement transducers (Pi-type) were set at the middle area of 100mm inlength on both sides to measure the crack width at the slit position. The loading speed was set to be 0.51mm/min as the head speed. Visible cracks observation and photographing were done after loading. In
addition, the upper and lower parts of the specimen were forcibly pulled apart and fibers on the fracturesurface were counted.

Figure 2.3 Setup of loading and measurement for uniaxial tension test10

2.3 Experiment Results
2.3.1 Compressive Properties
Before uniaxial tension test, compressive test was conducted to confirm the compressive strength andelastic modulus. For each series, three cylinder specimens (100 × 200mm) were tested. Table 2.4 lists
the compressive properties of each series. As shown by the results, the exact compressive strength wasmuch stronger than the target compressive strength. Compressive strength of the three mix proportions
showed obvious difference to each other. In addition, compressive strength decreased when fibers were mixed
into the matrix.

Series

Fc24

Fc36

Fc48

Table 2.4 Compressive properties (Uniaxial tension test)
Fiber volume fraction
Curing timeCompressive strength
Elastic modulus[%][Days]
[MPa]
[GPa]
None
38.3
14.2
1
25
35.6
13.6
2
33.3
13.0
None
54.6
18.2
1
32
48.2
17.8
2
46.1
17.6
None
71.4
20.0
1
45
66.9
19.7
2
63.7
19.3

11

2.3.2 Failure Mode under Tension
Figure 2.4 shows the failure modes of specimen under tension. Failure modes of specimens can bemainly divided into two types: tensile failure and bending failure. The specimens of tensile failure
generated one obvious crack throughout the slits on both sides. The specimens of bending failure
generated a slanting crack from one slit, and it did not penetrate to the slit at the other side. The occurof bending failure is considered to be caused by the nonuniform distribution of the fibers in the matrix.
The specimens, in which failure mode was detected to be bending failure, are not discussed in thefollowing parts.

(a)(b)
Figure 2.4 Failure modes of specimen under tension:(a) Tensile failure; (b) Bending failure

122.3.3 Crack Pattern and Crack Width
Figure 2.5 shows the examples of crack pattern of specimens in tensile failure. As revealed by the
figure, specimens without fibers (mortar) only generated one crack throughout the slits. As for DFRCC
specimens, an obvious crack throughout slits could be observed while multiple fine cracks generated
near the slits. Table 2.5 shows the number of cracks in testing area for each specimen. As for DFRCC
series, average number of cracks increased with increasing of fiber volume fraction and matrix strength.
In the gauge length of 100mm, the total crack width w was obtained by averaging the two values (w1,w2) measured by the two displacement transducers (Pi-type) on both sides of specimen ignoring theelastic deformation of mortar matrix.

13

N (Mortar)

1%

Fc24

Fc36

Fc48

Figure 2.5 Examples of crack pattern (tensile failure)

142%

Specimen
SeriesNo.
1
Fc24-N2
1
2
Fc24-1%
3
4
5
2
Fc24-2%
3
4
1
Fc36-N2
3
1
2
Fc36-1%
3
4
5
Fc36-2%4
2
Fc48-N3
4
1
2
Fc48-1%3
4
5
2
Fc48-2%
3
4

Table 2.5 Number of cracks in testing area
Number of cracks
Front side
Back side
Average
1
1
1.0
1
1
1.0
2
2
2.0
3
2
2.5
1
2
1.5
1
1
1.0
1
1
1.0
3
3
3.0
4
3
3.5
3
3
3.0
1
1
1.0
1
1
1.0
1
1
1.0
1
3
2.0
3
2
2.5
5
4
4.5
3
3
3.0
3
4
3.5
4
3
3.5
1
1
1.0
1
1
1.0
1
1
1.0
4
5
4.5
3
4
3.5
3
2
2.5
4
4
4.0
3
3
3.0
6
4
5.0
5
5
5.0
4
4
4.0

15

Average number ofcracks for each series
1.0

1.6

3.2

1.0

3.1

3.5
1.0

3.5

4.7

2.3.4 Load P - Crack width w Relationship
Figure 2.6 shows the load P - total crack width w relationship obtained from uniaxial tension test. In
order to compare the tensile performance between each series, the average curve of test results in eachseries is also shown in Figure 2.6 as red line. In the case of specimens without fibers (mortar), loaddecreased instantaneously to 0 when first crack generated. As for DFRCC specimens, load decreased
gradually after peak to a wide crack width. In addition, DFRCC specimens showed a tensile strainhardening property that load increases after the crack generated.
Table 2.6 shows the load at first crack P1 and crack width at first crack w1 for each specimen. Figure
2.7 shows the relationship between the load at first crack P1 and matrix strength fc. The lines in Figure
2.7 are connected by the average values of the load at first crack P1 in each series. As revealed by the
results, the load at first crack P1 of DFRCC specimens is larger than that of specimens without fibers(mortar), indicating that the addition of fibers has an inhibitory effect on crack occurring. The load atfirst crack P1 increased with increasing of fiber volume fraction and matrix strength.
Table 2.7 shows the maximum load Pmax and total crack width at maximum load wmax for eachspecimen. Figure 2.8 shows relationship between the maximum load Pmax and matrix strength fc. Thelines in Figure 2.8 are connected by the average values of the maximum load Pmax in each series. As
revealed by the results, the maximum load Pmax of DFRCC specimens is larger than the load at first crack
P1, which confirms a tensile strain-hardening property. By comparing the specimens with the same fiber
volume fraction, average maximum load Pmax.a increased as increasing of matrix strength. On the other
hand, by comparing the specimens with the same mix proportion of mortar matrix, average maximum
load Pmax.a increased as increasing of fiber volume fraction.

16

5

0.2
0.4
0.6
0.8
Total crack width w [mm]10Load P [kN]

Load P [kN]

10
5

0

0.2
0.4
0.6
0.8
Total crack width w [mm]

1

10

0.2
0.4
0.6
0.8
Total crack width w [mm]

1

2
4
6
8
Total crack width w [mm]Fc36-2%-4
15
10

0

10

2
4
6
8
Total crack width w [mm]1020Fc48-1%-1
Fc48-1%-2
Fc48-1%-3
Fc48-1%-4
Fc48-1%-5
Average

15
10

0

10

5

5

5

2
4
6
8
Total crack width w [mm]20Fc36-1%-1
Fc36-1%-2
Fc36-1%-3
Fc36-1%-4
Fc36-1%-5
Average

15

0

Load P [kN]

Load P [kN]

0

10

20
Fc48-N-2
Fc48-N-3
Fc48-N-4

10

0

2
4
6
8
Total crack width w [mm]

5

20
15

5

20
Fc36-N-1
Fc36-N-2
Fc36-N-3

Fc24-2%-2
Fc24-2%-3
Fc24-2%-4
Average

10

5

20
15

10

15

Load P [kN]0

15

Load P [kN]10

Fc24-1%-1
Fc24-1%-2
Fc24-1%-3
Fc24-1%-4
Fc24-1%-5
Average

Load P [kN]

15

20

20
Fc24-N-1
Fc24-N-2

Load P [kN]

Load P [kN]20

15

Fc48-2%-2
Fc48-2%-3
Fc48-2%-4
Average

10
5

2
4
6
8
Total crack width w [mm]100Figure 2.6 Load P – total crack width w relationship

17

2
4
6
8
Total crack width w [mm]10Table 2.6 Load at first crack P1 and crack width at first crack w1
Specimen
Load at first crack
Crack width at first crack
Average load at first crack
P1 [kN]w1 [mm]P1.a [kN]
Series
No.
1
4.99
0.011
Fc24-N
4.61
2
4.24
0.015
1
6.57
0.022
2
5.23
0.028
Fc24-1%
3
6.58
0.013
6.03
4
6.37
0.027
5
5.38
0.025
2
8.66
0.026
Fc24-2%
3
7.54
0.019
7.87
4
7.40
0.024
1
5.35
0.013
Fc36-N
2
5.40
0.014
5.35
3
5.30
0.014
1
7.51
0.014
2
9.53
0.023
Fc36-1%
3
8.10
0.020
8.36
4
7.69
0.023
5
8.94
0.017
Fc36-2%
4
10.34
0.018
10.34
2
6.30
0.014
Fc48-N
3
8.51
0.016
6.89
4
5.87
0.019
1
11.03
0.020
2
9.20
0.020
Fc48-1%
3
11.54
0.017
10.29
4
11.97
0.017
5
7.71
0.016
2
10.71
0.018
Fc48-2%
3
12.54
0.019
11.27
4
10.55
0.014

18

Table 2.7 Maximum load Pmax and total crack width at maximum load wmax
Specimen
Maximum load Total crack width at maximum load Average maximum load
Pmax [kN]
wmax [mm]Pmax.a [kN]
Series
No.
1
4.99
0.011
Fc24-N
4.61
2
4.24
0.015
1
8.88
0.780
2
8.54
1.106
Fc24-1%
3
9.76
0.627
8.99
4
8.49
1.100
5
9.28
0.750
2
13.33
0.241
Fc24-2%
3
12.05
0.559
12.60
4
12.41
0.466
1
5.35
0.013
Fc36-N
2
5.40
0.014
5.35
3
5.30
0.014
1
9.75
0.604
2
9.53
0.023
Fc36-1%
3
11.95
0.539
10.62
4
11.21
1.109
5
10.65
1.230
Fc36-2%
4
15.10
0.927
15.10
2
6.30
0.014
Fc48-N
3
8.51
0.016
6.89
4
5.87
0.019
1
13.25
0.828
2
12.82
0.824
Fc48-1%
3
11.80
1.275
12.63
4
11.97
0.017
5
13.33
0.500
2
18.28
0.865
Fc48-2%
3
17.27
1.391
17.03
4
15.53
0.327

19

Load at first crack P1 [kN]20
None fiber
1% fibers
2% fibers

15
10
5
0
30

40
50
60
70
Matrix strength fc [MPa]80

Figure 2.7 Relationship between load at first crack P1 and matrix strength fc

Maximum load Pmax [kN]

20
15
10
5
0
30

None fiber
1% fibers
2% fibers
40
50
60
70
Matrix strength fc [MPa]80

Figure 2.8 Relationship between maximum load Pmax and matrix strength fc

20

2.3.5 Fibers on Fracture Surface
Figure 2.9 shows the examples of the visual appearance of fracture surface in Fc36 series. It could
be detected that several fibers on the fracture surface unraveled when they were pulled out from themortar matrix. However, it is difficult to confirm whether the fibers ruptured or not due to tensile stressby visible observation.
Table 2.8 shows the number of fibers on fracture surface for each specimen counted after loading. As
revealed by the results, the specimen with a larger number of fibers showed a higher maximum load
Pmax. However, the average number of fibers on the fracture surface was not proportional as expected.
Although the measured amount of fibers was added at the mixture, inconstant fiber distribution could
be observed in the case of relatively small dimensions of specimen.

(a)(b)
Figure 2.9 Examples of the visual appearance of fracture surface (Fc36)(a) Fiber volume fraction: 1%; (b) Fiber volume fraction: 2%

21

Specimen
SeriesNo.
1
2
Fc24-1%
3
4
5
2
Fc24-2%3
4
1
2
Fc36-1%
3
4
5
Fc36-2%
4
1
2
Fc48-1%3
4
5
2
Fc48-2%
3
4

Table 2.8 Number of fibers on fracture surface
Number of fibers
Maximum load
Pmax [kN]
Top side
Bottom side
Total
8.88
99
106
205
8.54
90
108
198
9.76
109
112
221
8.49
95
97
192
9.28
103
109
212
13.33
182
151
333
12.05
159
145
304
12.41
174
129
303
9.75
96
82
178
9.53
89
79
168
11.95
118
109
227
11.21
107
93
200
10.65
101
95
196
15.10
163
152
315
13.25
101
97
198
12.82
105
86
191
11.80
92
94
186
11.97
97
101
198
13.33
107
99
206
18.28
156
148
304
17.27
163
144
307
15.53
131
149
280

22

Average numberof fibers

205.6

313.3

193.8

315.0

195.8

297.0

Chapter 3 Calculation of Bridging Law and Modeling
3.1 Introduction
In this chapter, theoretical bridging law is calculated based on the pullout characteristic of bundledaramid fiber obtained from previous study [10]. A coefficient α is introduced in the calculation
parameters to reflect the influence of matrix strength on bridging law. Coefficient α is varied to adapt
the calculation results with the experimental ones. The calculated bridging law is simplified to a bilinear
model, in which feature points are expressed by the function of fiber orientation intensity k and matrix
strength fc. After that, the modeled bridging law is applied to section analysis under bending condition.
The analyzed bending moment M - curvature φ relationships are compared with the results obtained
from 4-point bending test to verify the adaptability of modeled bridging law.

23

3.2 Calculation of Bridging Law
3.2.1 Calculation Method
The bridging law (bridging stress σbridge - crack width w relationship) is calculated by the method
proposed in previous study [10]. The bridging law can be obtained by the summation of forces carried
by individual bridging fibers considering the probability density function (PDF) for the fiber inclination
angle and the fiber centroidal location as given in Equation (1).

 bridge =Pbridge
Am

=

Vf
Af

  Pij ( w )  pxy (i )  pzx ( j )  px ( yh  zh )      ( y  z )
h

j

(1)

i

where,
σbridge = bridging stress,
Pbridge = bridging force (= total of pullout load),
Am = cross-sectional area of matrix,
Vf = fiber volume fraction,
Af = cross-sectional area of an individual fiber,P(w,ψ) = pullout load of an individual fiber,
pxy, pzx = probability density function for fiber inclination angle,
px = probability density function for fiber centroidal location,
ψ = fiber inclination angle to x-axis (= max{θ, ϕ}),θ = angle between x-axis and projected line of the fiber to x-y plane,
ϕ = angle between x-axis and projected line of the fiber to z-x plane,w = crack width.
The bilinear model proposed in previous study [10] is adapted for the pullout load of an individual
fiber, P(w,ψ). The elliptic distribution [12] is adopted for the probability density function (PDF), pxy andpzx, for fiber inclination angles. The elliptic distribution is defined by two parameters: principal
orientation angle θr (argument of one radius of elliptic function), and orientation intensity k (ratio of thetwo radii of elliptic function). The fiber orientation in mortar matrix can be expressed by these 2parameters. The random orientation is given by k = 1. When the value of k is larger than 1, fibers tend
to orient towards θr. On the other hand, when the value of k is smaller than 1, fibers tend to orient
perpendicular to θr. The PDF for fiber centroidal location px is set to be constant, which means the fibers
are randomly distributed along the longitudinal direction of the specimen.
The bridging stress calculated in this method do not include the tensile stress carried by the matrix
before cracking. In other words, the calculated bridging law exhibit the tensile stress due to only bridging
force of fibers.

24

3.2.2 Calculation Parameters
Table 3.1 shows the parameters for the calculation of bridging law. The calculation is conducted using
a bilinear model for pullout load - crack width relationship of an individual fiber proposed in previous
study [10]. In addition, inclined fiber angle and rupture of fiber are also considered in the calculation.
The orientation intensity k in the elliptic distribution for the PDF of the fiber inclination angel is set tobe 1.5 and 6 for the two planes parallel to the axial direction, which refers to previous study [12].
Furthermore, the principal orientation angle θr is set to be 0.
In the bilinear model of pullout behavior of an individual fiber, a coefficient α is introduced in theformula of the maximum pullout load Pmax to reflect the influence of matrix strength. In the calculationof bridging law, coefficient α is varied to adapt the calculation results with the experimental results.
Table 3.1 Parameters for the calculation of bridging law
Parameter
Cross-sectional area of an individual fiber Af [mm2]Length of fiber lf [mm]Apparent rupture strength of fiber σfu [MPa]

Input
0.196
30

 fu = 1080  e−0.667

Maximum pullout load Pmax [N]Pmax =   lb

Crack width at Pmax wmax [mm]wmax = 0.13  lb 0.64

Orientation intensity for x-y plane kxy
Orientation intensity for z-x plane kzx
Principle orientation angle θr [deg.]

1.5
6
0

Bilinear model

Elliptic distribution

where,
ψ = fiber inclination angle to x-axis [rad.],lb = embedded length of fiber [mm],α = coefficient reflecting the influence of matrix strength.

25

3.2.3 Calculation Results
From Section 2.3.5, the number of fibers on fracture surface was not observed to be proportional as
expected. In addition, the maximum loads Pmax in each series showed dispersion. In order to comparewith the calculated bridging law, the tensile load P is divided by the total number of fibers of eachspecimen to express the average tensile force carried by an individual fiber. In the case of calculated
bridging law, the tensile load (= Pbridge) is divided by the theoretical fiber number across the crack, which
can be calculated by Vf · Am / Af · ηf, where ηf, is the fiber effectiveness [12] at the crack width of 0.
When the orientation intensity kxy and kzx is 1.5 and 6 respectively, the fiber effectiveness ηf is 0.544.
According to Section 2.3.3, multiple cracks were detected in the gauge length of the displacementtransducers. Since the calculation of bridging law is performed on the fibers through only one cracking
surface, in order to compare the experimental results with the calculation ones, the experimental total
crack width w is corrected to be regarded as crack width through one crack. The experimental value of
total crack width w up to the last peak of the curves is divided by the average number of cracks to obtaina corrected crack width through one crack. The average number of cracks in the gauge length for eachspecimen is obtained from Section 2.3.3. After the last peak, the crack opening is considered to beconcentrated in one crack, so the post-peak branch of the curves is considering to be the same as theexperimental results.
Table 3.2 shows the maximum load per an individual fiber Pind.max for each specimen and calculation
result. Figure 3.1 shows the tensile load per fiber Pind. - crack width w relationship, in which black lines
are the corrected experimental results, while red lines are the calculation results. As revealed by the
results, the softening branch of the calculated bridging law shows good agreement with the experimental
results.
As mentioned in Section 3.2.2, the coefficient α, which reflects the influence of matrix strength, isvaried to adapt the calculation results with the experimental ones. Each series of specimens corresponds
to a coefficient α, which is shown in Table 3.2 and Figure 3.1. Linked with the matrix strength fc shownin section 2.3.1, Figure 3.2 shows the relationship between coefficient α and matrix strength fc. The
black lines in Figure 3.2 exhibit the modified regression calculation results by the least square method,
to simplify the relational expression between coefficient α and matrix strength fc, as given by Equation(2).

 = 0.16  f c
where,
unit for matrix strength fc is MPa.

26

(2)

Table 3.2 (i) Maximum load per an individual fiber Pind.max (Fc24)
Specimen
Number of fibers
Maximum load
Maximum load per
Pmax [kN]fiber Pind.max [N]SeriesNo.
Top side
Bottom side
Total
1
8.88
99
106
205
43.33
2
8.54
90
108
198
43.13
Fc24-1%
3
9.76
109
112
221
44.15
4
8.49
95
97
192
44.23
5
9.28
103
109
212
43.76
Cal.

5.03

Fc24-2%

2
3
4

13.33
12.05
12.41

Fc24-2%(α=5.5)

Cal.

9.29

= V f  A m / A f  f
= 0.01 ( 60  70 ) / 0.196  0.544
182
159
174

151
145
129

= V f  A m / A f  f
= 0.02  ( 60  70 ) / 0.196  0.544

70

Tensile load per fiber Pind. [N]Tensile load per fiber Pind. [N]Fc24-1%(α=6.0)

Fc24-1%-Exp.
Fc24-1%-Cal.
(=)

60
50
40
30
20
10
0

2

4
6
8
Crack width w [mm]

43.23

333
304
303

40.04
39.63
40.96

233.1

39.91

70
Fc24-2%-Exp.
Fc24-2%-Cal.
(=)

60
50
40
30
20
10
0

10

116.6

2

4
6
8
Crack width w [mm]Figure 3.1 (i) Tensile load per fiber Pind. - crack width w relationship (Fc24)

27

10

Table 3.2 (ii) Maximum load per an individual fiber Pind.max (Fc36)
Specimen
Number of fibers
Maximum load
Maximum load per
Pmax (kN)fiber Pind.max [N]SeriesNo.
Top side
Bottom side
Total
1
9.75
96
82
178
54.76
2
9.53
89
79
168
56.73
Fc36-1%
3
11.95
118
109
227
52.63
4
11.21
107
93
200
56.07
5
10.65
101
95
196
54.33
Cal.

6.31

Fc36-2%

4

15.10

Fc36-2%(α=6.8)

Cal.

11.15

= V f  A m / A f  f
= 0.01 ( 60  70 ) / 0.196  0.544
163

152

= V f  A m / A f  f
= 0.02  ( 60  70 ) / 0.196  0.544

70

Tensile load per fiber Pind. [N]Tensile load per fiber Pind. [N]Fc36-1%(α=8.2)

Fc36-1%-Exp.
Fc36-1%-Cal.
(=)

60
50
40
30
20
10
0

2

4
6
8
Crack width w [mm]

54.22

315

47.92

233.1

47.92

70
Fc36-2%-Exp.
Fc36-2%-Cal.
(=)

60
50
40
30
20
10
0

10

116.6

2

4
6
8
Crack width w [mm]Figure 3.1 (ii) Tensile load per fiber Pind. - crack width w relationship (Fc36)

28

10

Table 3.2 (iii) Maximum load per an individual fiber Pind.max (Fc48)
Specimen
Number of fibers
Maximum load
Maximum load per
Pmax (kN)fiber Pind.max [N]SeriesNo.
Top side
Bottom side
Total
1
13.25
101
97
198
66.91
2
12.82
105
86
191
67.13
Fc48-1%
3
11.80
92
94
186
63.44
4
11.97
97
101
198
60.46
5
13.33
107
99
206
64.72
Cal.

7.25

Fc48-2%

2
3
4

18.28
17.27
15.53

Fc48-2%(α=9.1)

Cal.

13.32

= V f  A m / A f  f
= 0.01 ( 60  70 ) / 0.196  0.544
156
163
131

148
144
149

= V f  A m / A f  f
= 0.02  ( 60  70 ) / 0.196  0.544

70

Tensile load per fiber Pind. [N]Tensile load per fiber Pind. [N]Fc48-1%(α=11.2)

Fc48-1%-Exp.
Fc48-1%-Cal.
(=)

60
50
40
30
20
10
0

2

4
6
8
Crack width w [mm]62.320

304
307
280

60.14
56.25
55.46

233.1

57.24870
Fc48-2%-Exp.
Fc48-2%-Cal.
(=)

60
50
40
30
20
10
0

10

116.6

2

4
6
8
Crack width w [mm]Figure 3.1 (iii) Tensile load per fiber Pind. - crack width w relationship (Fc48)

29

10

20

Coefficient 

0.16 · fc
15
10
5

0

20
40
60
Matrix strength fc [MPa]80

Figure 3.2 Relationship between coefficient α and matrix strength fc

30

3.3 Modeling of Bridging Law

Bridging stress bridge [MPa]The calculated bridging law, in which matrix strength fc is set to be 45MPa, for the orientation
intensity k ranging from 0.1 to 10 are shown in Figure 3.3. In order to simply express the elliptic
distribution, orientation intensity k, which is the average value of kxy and kzx, is applied in the modeling.
The bridging laws shown in the figure are calculated with 0.1 intervals of k in the case of k < 1, and with
1 interval when k > 1. The parameters adopted for the calculation are set the same as those in Table 3.1,
excepting fiber orientation intensity k. The coefficient α are calculated by the relational expression withmatrix strength fc, as given by Equation (2). In addition, volume fraction of fibers is set to be 2%. As
revealed by the results, bridging stress σbridge decreases gradually after peak, and becomes 0 when crack
width reaches to 15mm (half of fiber length) because most of the fibers are completely pulled out fromthe matrix. On the other hand, by comparing each curve, the maximum bridging stress remarkably
increases with the increase of orientation intensity k, which means that bridging stress σbridge becomeslarger when the fibers strongly orient to the normal direction of the crack surface.

5
k=0.1
k=1
k=10

4
3
2
1
0

5
10
Crack width w [mm]15

Figure 3.3 Calculation results of bridging law (fc = 45MPa)
The calculated bridging laws are modeled by simple forms considering fiber orientation and matrix
strength, to utilize them effectively in various types of structural elements. As shown in Figure 3.4,
bridging law is simply characterized by two regions: the curve until the maximum stress and softening
branch, so the bridging law is simplified to a bilinear model. The bilinear model has three parameters:
the maximum bridging stress σmax, the crack width at maximum bridging stress wmax, and the crack width
when bridging stress reach 0 wtu. The values of σmax and wmax of the model can be obtained directly fromthe calculation results. The value of wtu is determined to have an equivalent fracture energy with thecalculated bridging law in the softening branch. The modeled bridging law, in which matrix strength fc
is set to be 45MPa, for the orientation intensity k ranging from 0.1 to 10 are shown in Figure 3.5.

31

Bridging stress bridge [MPa]Figure 3.4 Bilinear model for bridging law

5
k=0.1
k=1
k=10

4
3
2
1
0

5
10
Crack width w [mm]15

Figure 3.5 Modeled bridging law (fc = 45MPa)
Figure 3.6 shows the relationship between each parameter of bilinear model and orientation intensityk (fc = 45MPa). The black line in each figure exhibits the modified regression calculation results by theleast square method, to simplify the relational expression between each parameter and fiber orientation
intensity k. In order to take the influence of matrix strength into consideration, it is assumed that eachparameter can be expressed as the functions given by Equations (3) to (5). Coefficients A to E in theequations are considered to be affected by matrix strength fc.

 max = A  k B

 MPa (3)

wmax = C  k D

 mm

(4)

wtu = E  k F

 mm

(5)

32

5

1
1.87 · k

0.33

0.60 · k
0.8

wmax [mm]max [MPa]

4
3
2

0.6
0.4
0.2

1
0

0.065

0

5
Orientation intensity k

0

10

0

5
Orientation intensity k

(a)

10

(b)
15

wtu [mm]

9.46 · k

0.045

10

5

0

0

5
Orientation intensity k

10(c)
Figure 3.6 Relationship between each parameter and orientation intensity k (fc = 45MPa):(a) Maximum bridging stress σmax; (b) Crack width at maximum bridging stress wmax;(c) Crack width when bridging stress reach 0 wtu
The same calculations are conducted with 5MPa intervals of matrix strength fc in the case of 20MPa
to 60MPa, as shown in Figure 3.7. According to the calculation for each matrix strength fc, coefficients
A to E can be obtained from the modified regression calculation results of each parameter, as shown in
Figure 3.8. Table 3.3 shows the coefficients of parameters for bilinear model in each matrix strength fc.
However, when matrix strength fc is smaller than 30MPa, the calculation results of k ranging from 0.1
to 10 do not show any rupture of fibers, as shown in Figure 3.8 and Figure 3.9. Consequently, wmax andwtu do not change obviously with the change of fiber orientation intensity k, as shown in Figure 3.8.
Therefore, when matrix strength fc is smaller than 30MPa, the values of wmax and wtu are obtained fromthe average values of calculation results.

33

Bridging stress bridge [MPa]Bridging stress bridge [MPa]

5
k=0.1
k=1
k=10

4
3
2
1
0

5
10
Crack width w [mm]

5
k=0.1
k=1
k=10

4
3
2
1
0

15

5
10
Crack width w [mm]

15

(a)
(b)
Figure 3.7 (i) Calculated and modeled bridging law (fc = 20MPa)(a) Calculation results; (b) Modeled bridging law
5

10.90 · k

0.30

0.72
0.8

wmax [mm]max [MPa]

4
3
2

0.4
0.2

1
0

0.6

0

5
Orientation intensity k

0

10

0

5
Orientation intensity k

(a)

10

(b)
15

wtu [mm]

10.37
10

5

0

0

5
Orientation intensity k

10(c)
Figure 3.8 (i) Relationship between each parameter and orientation intensity k (fc = 20MPa):(a) Maximum bridging stress σmax; (b) Crack width at maximum bridging stress wmax;(c) Crack width when bridging stress reach 0 wtu

34

Bridging stress bridge [MPa]Bridging stress bridge [MPa]

5
k=0.1
k=1
k=10

4
3
2
1
0

5
10
Crack width w [mm]

5
k=0.1
k=1
k=10

4
3
2
1
0

15

5
10
Crack width w [mm]

15

(a)
(b)
Figure 3.7 (ii) Calculated and modeled bridging law (fc = 25MPa)(a) Calculation results; (b) Modeled bridging law
5

1
1.12 · k

0.30

0.72
0.8

wmax [mm]max [MPa]

4
3
2

0.4
0.2

1
0

0.6

0

5
Orientation intensity k

0

10

0

5
Orientation intensity k

(a)

10

(b)
15

wtu [mm]

10.37
10

5

0

0

5
Orientation intensity k

10(c)
Figure 3.8 (ii) Relationship between each parameter and orientation intensity k (fc = 25MPa):(a) Maximum bridging stress σmax; (b) Crack width at maximum bridging stress wmax;(c) Crack width when bridging stress reach 0 wtu

35

Bridging stress bridge [MPa]Bridging stress bridge [MPa]

5
k=0.1
k=1
k=10

4
3
2
1
0

5
10
Crack width w [mm]

5
k=0.1
k=1
k=10

4
3
2
1
0

15

5
10
Crack width w [mm]

15

(a)
(b)
Figure 3.7 (iii) Calculated and modeled bridging law (fc = 30MPa)(a) Calculation results; (b) Modeled bridging law
51
1.35 · k

0.30

0.72
0.8

wmax [mm]max [MPa]

4
3
2

0.4
0.2

1
0

0.6

0

5
Orientation intensity k

0

10

0

5
Orientation intensity k

(a)

10

(b)
15

wtu [mm]

10.37
10

5

0

0

5
Orientation intensity k

10(c)
Figure 3.8 (iii) Relationship between each parameter and orientation intensity k (fc = 30MPa):(a) Maximum bridging stress σmax; (b) Crack width at maximum bridging stress wmax;(c) Crack width when bridging stress reach 0 wtu

36

Bridging stress bridge [MPa]Bridging stress bridge [MPa]

5
k=0.1
k=1
k=10

4
3
2
1
0

5
10
Crack width w [mm]

5
k=0.1
k=1
k=10

4
3
2
1
0

15

5
10
Crack width w [mm]

15

(a)
(b)
Figure 3.7 (iv) Calculated and modeled bridging law (fc = 35MPa)(a) Calculation results; (b) Modeled bridging law
51
1.56 · k

0.68 · k

0.31

0.8

wmax [mm]max [MPa]

4
3
2

0.6
0.4
0.2

1
0

0.026

0

5
Orientation intensity k

0

10

0

5
Orientation intensity k

(a)

10

(b)
15

wtu [mm]10.17 · k

0.0085

10

5

0

0

5
Orientation intensity k

10(c)
Figure 3.8 (iv) Relationship between each parameter and orientation intensity k (fc = 35MPa):(a) Maximum bridging stress σmax; (b) Crack width at maximum bridging stress wmax;(c) Crack width when bridging stress reach 0 wtu

37Bridging stress bridge [MPa]Bridging stress bridge [MPa]

5
k=0.1
k=1
k=10

4
3
2
1
0

5
10
Crack width w [mm]

5
k=0.1
k=1
k=10

4
3
2
1
0

15

5
10
Crack width w [mm]

15

(a)
(b)
Figure 3.7 (v) Calculated and modeled bridging law (fc = 40MPa)(a) Calculation results; (b) Modeled bridging law
51
1.73 · k

0.32

0.64 · k
0.8

wmax [mm]max [MPa]

4
3
2

0.6
0.4
0.2

1
0

0.050

0

5
Orientation intensity k

0

10

0

5
Orientation intensity k

(a)

10

(b)
15

wtu [mm]

9.88 · k

0.024

10

5

0

0

5
Orientation intensity k

10(c)
Figure 3.8 (v) Relationship between each parameter and orientation intensity k (fc = 40MPa):(a) Maximum bridging stress σmax; (b) Crack width at maximum bridging stress wmax;(c) Crack width when bridging stress reach 0 wtu

38

Bridging stress bridge [MPa]Bridging stress bridge [MPa]

5
k=0.1
k=1
k=10

4
3
2
1
0

5
10
Crack width w [mm]

5
k=0.1
k=1
k=10

4
3
2
1
0

15

5
10
Crack width w [mm]

15

(a)
(b)
Figure 3.7 (vi) Calculated and modeled bridging law (fc = 45MPa)(a) Calculation results; (b) Modeled bridging law
51
1.87 · k

0.33

0.60 · k
0.8

wmax [mm]max [MPa]

4
3
2

0.6
0.4
0.2

1
0

0.065

0

5
Orientation intensity k

0

10

0

5
Orientation intensity k

(a)

10

(b)
15

wtu [mm]

9.46 · k

0.045

10

5

0

0

5
Orientation intensity k

10(c)
Figure 3.8 (vi) Relationship between each parameter and orientation intensity k (fc = 45MPa):(a) Maximum bridging stress σmax; (b) Crack width at maximum bridging stress wmax;(c) Crack width when bridging stress reach 0 wtu

39Bridging stress bridge [MPa]Bridging stress bridge [MPa]

5
k=0.1
k=1
k=10

4
3
2
1
0

5
10
Crack width w [mm]

5
k=0.1
k=1
k=10

4
3
2
1
0

15

5
10
Crack width w [mm]

15

(a)
(b)
Figure 3.7 (vii) Calculated and modeled bridging law (fc = 50MPa)(a) Calculation results; (b) Modeled bridging law
510.56 · k
0.8

wmax [mm]max [MPa]

4
3
2
1
0

0.079

1.98 · k05Orientation intensity k

0.6
0.4
0.2

0.34

0

10

0

5
Orientation intensity k

(a)

10

(b)
15

wtu [mm]

8.92 · k

0.071

10

5

0

0

5
Orientation intensity k

10(c)
Figure 3.8 (vii) Relationship between each parameter and orientation intensity k (fc = 50MPa):(a) Maximum bridging stress σmax; (b) Crack width at maximum bridging stress wmax;(c) Crack width when bridging stress reach 0 wtu

40

Bridging stress bridge [MPa]Bridging stress bridge [MPa]

5
k=0.1
k=1
k=10

4
3
2
1
0

5
10
Crack width w [mm]

5
k=0.1
k=1
k=10

4
3
2
1
0

15

5
10
Crack width w [mm]

15

(a)
(b)
Figure 3.7 (viii) Calculated and modeled bridging law (fc = 55MPa)(a) Calculation results; (b) Modeled bridging law
510.52 · k
0.8

wmax [mm]max [MPa]

4
3
2
1
0

0.091

2.07 · k05Orientation intensity k

0.6
0.4
0.2

0.35

0

10

0

5
Orientation intensity k

(a)

10

(b)
15

wtu [mm]8.32 · k

0.097

10

5

0

0

5
Orientation intensity k

10(c)
Figure 3.8 (viii) Relationship between each parameter and orientation intensity k (fc = 55MPa):(a) Maximum bridging stress σmax; (b) Crack width at maximum bridging stress wmax;(c) Crack width when bridging stress reach 0 wtu

41

Bridging stress bridge [MPa]Bridging stress bridge [MPa]

5
k=0.1
k=1
k=10

4
3
2
1
0

5
10
Crack width w [mm]

5
k=0.1
k=1
k=10

4
3
2
1
0

15

5
10
Crack width w [mm]

15

(a)
(b)
Figure 3.7 (ix) Calculated and modeled bridging law (fc = 60MPa)(a) Calculation results; (b) Modeled bridging law
510.49 · k
0.8

wmax [mm]max [MPa]

4
3
2
1
0

0.093

2.15 · k05Orientation intensity k

0.6
0.4
0.2

0.35

0

10

0

5
Orientation intensity k

(a)

10

(b)
15

wtu [mm]

7.67 · k

0.12

10

5

0

0

5
Orientation intensity k

10(c)
Figure 3.8 (ix) Relationship between each parameter and orientation intensity k (fc = 60MPa):(a) Maximum bridging stress σmax; (b) Crack width at maximum bridging stress wmax;(c) Crack width when bridging stress reach 0 wtu

425Bridging stress bridge [MPa]Bridging stress bridge [MPa]Table 3.3 Coefficients of parameters for bilinear model
Coefficients of parameters for bilinear model
Matrix strength fc
[MPa]
A
B
C
D
E
20
0.90
0.30
0.72*
0
10.37*
25
1.12
0.30
0.72*
0
10.37*
30
1.35
0.30
0.72*
0
10.37*
35
1.56
0.31
0.68
0.026
10.17
40
1.73
0.32
0.64
0.050
9.88
45
1.87
0.33
0.60
0.065
9.46
50
1.98
0.34
0.56
0.079
8.92
55
2.07
0.35
0.52
0.091
8.32
60
2.15
0.35
0.49
0.093
7.67
* Obtained by the average values of calculation results

fc = 30MPa k = 1
fc = 60MPa k = 1

4
3

Rupture of fibers

2
1
No rupture of fiber
0

1

2
3
4
Crack width w [mm]

5
k=0.1
k=1
k=10

4
3
2
1
0

5

F
0
0
0
0.0085
0.024
0.045
0.071
0.097
0.12

1

2
3
4
Crack width w [mm]

5

(a)
(b)
Figure 3.9 Calculation results of bridging law (fc ≤ 30MPa)(a) Bridging law showing rupture of fibers or not; (b) Calculated bridging law (fc = 30MPa)

43

Figure 3.10 shows the relationship between each coefficient and matrix strength fc. The black line ineach figure exhibits the modified regression calculation results by the least square method, to simplify
the relational expression between each coefficient and matrix strength fc. In the case of coefficient B toF, the solid lines exhibit the modified regression calculation results when fc is larger than 30MPa, while
dotted lines exhibit the ones when fc is smaller than 30MPa. The relational expression between eachcoefficient and matrix strength fc is given by Equations (6) to (11).

A = 0.087  f c 0.80

f c   20, 60

(6)

0.30
B=
0.24
0.13  f c

f c   20,30
f c  ( 30, 60

(7)

0.72
C=
0.95 − 0.0078  f c
0D=0.19 − 5.82 / f c

f c   20,30
f c  ( 30, 60

(8)f c   20,30
f c  ( 30, 60

10.37E=
2
9.65 + 0.082  f c − 0.0019  f c(9)f c   20,30
f c  ( 30, 60

0
F =
2
−0.012 − 0.0016  f c + 0.000064  f c

f c   20,30
f c  ( 30, 60

(10)

(11)

where,
unit for matrix strength fc is MPa.
The feature points of bilinear model of bridging law in each fiber orientation intensity k and matrix
strength fc, can be easily obtained using Equations (3) to (11).

44

0.5

3

Coefficient B

Coefficient A

0.4
2

1
0.087 · fc
0

20
40
60
Matrix strength fc [MPa]

0.3
0.2

0.30
( fc30MPa )0.24
0.13 · fc( fc>30MPa )

0.1

0.80

0

80

20
40
60
Matrix strength fc [MPa]

(a)

(b)
0.15

1

Coefficient D

Coefficient C

0.8
0.6
0.4
0.2
0

0.72
( fc30MPa )0.95 - 0.0078 · fc
( fc>30MPa )20
4060
Matrix strength fc [MPa]

0.1

0
0

80

Coefficient F

Coefficient E

80

0.15

10
8
6

0

20
40
60
Matrix strength fc [MPa]

(d)

12

2

0
( fc30MPa )0.19 - 5.82 / fc
( fc>30MPa )

0.05

(c)

4

80

10.37
( fc30MPa )2
9.65 + 0.082 · fc - 0.0019 · fc
( fc>30MPa )20
4060
Matrix strength fc [MPa]0.10( fc30MPa )-0.012 - 0.0016 · fc
2
+ 0.000064 · fc
( fc>30MPa )

0.05

0
0

80

20
40
60
Matrix strength fc [MPa](e)(f)
Figure 3.10 Relationship between each coefficient and matrix strength fc(a) Coefficient A; (b) Coefficient B; (c) Coefficient C;(d) Coefficient D; (e) Coefficient E; (f) Coefficient F

45

80

3.4 Adaptability of Modeled Bridging Law
3.4.1 4-point Bending Test
4-point bending test is conducted for DFRCC specimens to investigate the bending performanceexperimentally. Bending specimens are rectangular prisms of 100×100×400mm. The applied materials
are the same as those described in Section 2.2.2. For each series, five bending specimens were
manufactured and cured in the natural environment.
4-point bending test was carried out using a universal testing machine of 2MN capacity. Image of 4point bending test is shown in Figure 3.11. Load was applied to the specimen on the trisection points ofspan, and loading speed was set to be 0.5mm/min. Two displacement transducers (Pi-type) were set atthe compression and tension sides of constant bending moment area with a vertical distance of 70mm.
Measurement items were load P, axial deformation of compression side δ1 and axial deformation oftension side δ2. Bending moment M and curvature φ can be calculated by Equations (12) to (13) andbending moment M - curvature φ relationship can be obtained.

P L

2 3

 kN  m

(12)

 2 − 1d0

1/m 

(13)

M=

=

where,
P = load applied to specimen,
L = span (= 300mm),
ε1, ε2 = strain obtained from dividing deformation δ1 and δ2 by 100mm (gauge length)d0 = vertical distance between two displacement transducers (= 70mm)
The bending specimens, in which the localized crack opened out of the constant bending moment
area, are not discussed in the following parts.
Before 4-point bending test, compressive test was also conducted to confirm the compressive strengthand elastic modulus. For each series, three cylinder specimens (100 × 200mm) were tested. Table 3.4
shows the compressive properties of each series. As revealed by the results, compressive strength of thethree mix proportions showed obvious difference to each other. In addition, by comparing with results
shown in Table 2.4, compressive strength increased remarkedly with increasing of curing time.

46

Figure 3.11 Image of 4-point bending test

Series

Fc24

Fc36

Fc48

Table 3.4 Compressive properties (4-point bending test)
Fiber volume fraction
Curing timeCompressive strength
Elastic modulus[%][Days]
[MPa]
[GPa]
None
26.6
11.2
1
6
24.4
11.2
2
24.0
10.7
None
37.1
14.9
1
8
34.8
15.2
2
35.1
14.5
None
52.6
17.4
1
11
53.7
17.8
2
53.0
17.0

47

3.4.2 Section Analysis
In order to verify the adaptability of the modeled bridging law proposed in Section 3.3, the section
analysis under bending condition based on the modeled bridging law is conducted.
Figure 3.12 shows the stress σ - strain ε model applied in section analysis. As for compression side,a parabola model based on the experimental results of compressive test is used. As for tension side, thetrilinear model consists of two parts: elastic part and bridging law part. Feature point of elastic part isobtained based on experimental results, which parameters can be calculated by Equations (14) to (15).
Feature points of bridging law part are obtained from the bilinear model of bridging law proposed in
Section 3.3, which parameters can be calculated by Equations (16) to (18). Table 3.5 shows theparameters for the model applied in section analysis of each series. The fiber orientation intensity k is
set to be 3.8, which refers to previous study [12]. Matrix strength fc is obtained from the compressive
test results of cylinder specimens manufactured in the same batch of bending specimens. As for theseries of 1% fibers, it is assumed that stress σ in bridging law part is half of that of 2% fibers.

 cr =

M cr 6  M cr=Zb  h2

 cr =

 cr
Ec

 MPa (14)
(15)

where,
Mcr = bending moment when the first crack generates (obtained from 4-point bending test),
Z = section modulus (= b·h2/6),
b, h = dimensions of cross-section,
Ec = elastic modulus (obtained from compressive test).

 t =  max

 MPa 

(16)

t =

wmax
lt

(17)

u =

wtu
lt

(18)

where,
σmax, wmax, wtu = parameters of bilinear model of bridging law (Equation (3) to (5)),
lt = length of constant bending moment area of 4-point bending test (= 100mm).

48

The section analysis is conducted based on the assumption that cross-section remains plain when
considering its deformation. Firstly, curvature φ is given. After that, the strain ε of each element incross-section is calculated from linear distribution of strain and stress for each element is obtained fromthe stress σ - strain ε model shown in Figure 3.12. Finally, neutral axis satisfying equilibrium condition
is found numerically and bending moment M can be calculated. When calculations are conducted forcurvature φ from 0 to 0.8 (unit: 1/m) with fine intervals, analyzed bending moment M - curvature φrelationships can be obtained.

Figure 3.12 Stress σ - strain ε model applied in section analysis
Table 3.5 Parameters for stress σ - strain ε model applied in section analysis
Tension side
Compression
side
fc
Elastic part
Bridging law part
Series
k
[MPa]
σc
σcr
σt
εc
εcr
εt
εu
[MPa]
[MPa]
[MPa]
Fc24-1%24.4-24.4
2.136 0.00019 0.836 0.0072 0.104
Fc24-2%24.0-24.0
2.604 0.00024 1.651 0.0072 0.104
Fc36-1%34.8-34.8
2.772 0.00018 1.118
0.0070 0.103
3.8-0.005
Fc36-2%35.1-35.1
2.982 0.00021 2.253 0.0070 0.103
Fc48-1%53.7-53.7
2.946 0.00016 1.654 0.0059 0.096
Fc48-2%53.0-53.0
3.432 0.00020 3.269 0.0060 0.097

49

3.4.3 Comparison of Analysis and Experimental Results
Figure 3.13 shows the bending moment M - curvature φ relationship, in which black lines exhibit
the experimental results obtained from 4-point bending test, while red lines exhibit the analysis results.
According to the observation after loading, the number of cracks in constant bending moment area of
DFRCC specimens did not show obvious difference with the change of fiber volume fraction. Table
3.6 shows the maximum bending moment Mmax of experimental and analysis results for each series. As
revealed in Figure 3.13, analysis results show good agreements with experimental results in the seriesof 2% fiber volume fraction. As for the series of 1% fiber volume fraction, the difference of the shapeof post-peak branch is considered to be due to the number and distribution of fibers across the fracturesurface. As shown in Table 3.6, maximum bending moment Mmax of experimental results is from 0.910
to 1.105 times of that of analysis results.
Table 3.6 Maximum bending moment Mmax of experimental and analysis results
Maximum bending moment Mmax [kN·m]
SeriesExp. / Ana.
Exp.*
Ana.
Fc24-1%
0.714
0.723
0.988
Fc24-2%
0.925
0.906
1.021
Fc36-1%
1.048
0.948
1.105
Fc36-2%
1.207
1.104
1.093
Fc48-1%
1.051
1.065
0.987
Fc48-2%
1.303
1.432
0.910
* Average value of experimental results

50

1
0.5

0.2
0.4
0.6
Curvature  [1/m]2
1.5

Fc36-1%-Exp.
Fc36-1%-Ana.

1
0.5

0

0.2
0.4
0.6
Curvature  [1/m]Fc48-1%-Exp.
Fc48-1%-Ana.

1
0.5

0

0.2
0.4
0.6
Curvature  [1/m]

1
0.5

0.2
0.4
0.6
Curvature  [1/m]Fc36-2%-Exp.
Fc36-2%-Ana.

1.5
1
0.5

0.2
0.4
0.6
Curvature  [1/m]0.8

2
Fc48-2%-Exp.
Fc48-2%-Ana.

1.5
1
0.5

0.2
0.4
0.6
Curvature  [1/m]Figure 3.13 Bending moment M - curvature φ relationship

51

0.8

2

0

0.8

Fc24-2%-Exp.
Fc24-2%-Ana.

1.5

0

0.8

2
1.5

2

0

0.8

Bending moment M [kN·m]

Bending moment M [kN·m]

0

Bending moment M [kN·m]

Bending moment M [kN·m]

1.5

Fc24-1%-Exp.
Fc24-1%-Ana.

Bending moment M [kN·m]

Bending moment M [kN·m]

20.8

Chapter 4 Conclusion
This study investigated the influence of matrix strength on the tensile performance of DFRCC withbundled aramid fibers. A total of 45 specimens varying matrix strength and fiber volume fraction were
subjected to uniaxial tension test. Based on the experimental and calculation results, bridging law model,
in which feature points were expressed by the function of fiber orientation intensity and matrix strength
was proposed. Adaptability of the modeled bridging law was verified through section analysis for 4point bending test. The main conclusions of this study are summarized as follows.
(1) In uniaxial tension test, according to the observation after loading, most of the DFRCC specimens
in tensile failure showed multiple cracks. Number of cracks increased with increasing of matrix
strength. The load at first cracking and the maximum load also increased with increasing of matrix
strength.
(2) From the results of the calculation of bridging law, the modified individual fiber pullout model, in
which the maximum pullout force is given by the proportional relations with the compressive
strength of matrix, could express the uniaxial tension test results well.
(3) The calculated bridging law using the modified fiber pullout model was simplified to a bilinear
model, in which feature points were expressed by the function of fiber orientation intensity andmatrix strength.
(4) Section analysis under bending condition was conducted for series with different matrix strengthand fiber volume fraction using the modeled bridging law. The maximum bending moment ofexperimental results was from 0.910 to 1.105 times of that of analysis results.

52

Acknowledgements
First of all, please allow me to express my sincerest appreciation to my academic supervisor, Prof.
Toshiyuki Kanakubo, for his precious guidance, constructive supervision, priceless feedback, andpleasant communication during this research project. Without his encouragement and support, this thesis
could not be completed. His extensive research experience and keen insight have impressed me deeply
and will certainly encourage me for the rest of my life.
I would also like to convey my appreciation to my vice-academic supervisor Associate Prof. Akira
Yasojima and a technical staff member Mr. Kojima, for their extremely important assistance on myexperiment and their valuable comments on my research project.
In addition, I would like to thank all the members in this laboratory for their kind support and creatingsuch a nice working environment. Special thanks go to Mr. Daiki Sunaga, Mr. Takumi Koba, and Mr.
Jin Wang for their help in executing experimental works.
Finally, special thanks are also expressed to my dearest parents for their endless love and continuous
support in my life.

53

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