D

Journal of Civil Engineering and Architecture 12 (2018) 651-661
doi: 10.17265/1934-7359/2018.09.005

DAVID

PUBLISHING

Modeling of Bridging Law for PVA Fiber-Reinforced
Cementitious Composite Considering Fiber Orientation
Yuriko Ozu1, Masaru Miyaguchi1 and Toshiyuki Kanakubo2
1. Sumitomo Mitsui Construction Co., Ltd., Chuo-ku, Tokyo 104-0051, Japan
2. Department of Engineering Mechanics and Energy, University of Tsukuba, Tsukuba-city, Ibaraki 305-8573, Japan
Abstract: The authors have proposed the calculation method of bridging law, that is expressed by tensile stress–crack width
relationship, considering the influence of fiber orientation in FRCC (fiber-reinforced cementitious composite). The objective of thisstudy is to propose a new tri-linear model that expresses the bridging law considering fiber orientation. The parameters that give thecharacteristic points of the tri-linear model are proposed as functions of orientation intensity. The bending test, in which thespecimens are fabricated by three different casting methods, is conducted to verify the adaptability of the proposed model. The
results of section analysis using the proposed model can present the difference of bending strength due to the fiber orientation.
Key words: FRCC, tensile stress, crack width, tri-linear model, fiber orientation, PVA fiber, bending test, compacting vibrator.

1. Introduction
FRCC (fiber-reinforced cementitious composite), in
which short discrete fibers of a certain percentage involume fraction are mixed in mortar or concrete, iscementitious material that shows higher tensile andbending performance comparing with conventional
concrete. The elements such as coupling beams andseismic walls using SHCC (strain hardening cement
composites), that show tensile strain hardening andmultiple fine cracks, provide very ductile behavior
with small crack opening (e.g. [1]). These
characterized performances of FRCC are brought bybridging effect of fibers at cracks in the matrix.
However, some previous studies have reported thatthe tensile characteristics even in SHCC are
influenced by fiber orientation in matrix (e.g. [2]).
Casting and pouring direction of FRCC affects thefiber orientation, and vertical pouring in tension test
specimens causes degradation of tensile strength anddeformation capacity of FRCC.
Corresponding author: Toshiyuki Kanakubo, professor,
research fields: concrete structures, cementitious composites,
FRP for concrete strengthening and seismic performance ofconcrete structures.

The authors have studied the influence of fiber
orientation to tensile characteristics of FRCC using
PVA (polyvinyl alcohol) fiber through visualization
simulation using water glass solution and calculationof the bridging law, which is expressed by tensilestress–crack width relationship [3]. To evaluate thefiber orientation distribution quantitatively, anapproximation methodology using an elliptic function(elliptic distribution) was introduced in that study. The
bridging law is calculated considering the elliptic
distribution, the snubbing effect [4], and the fiber
strength degradation [5]. The calculated bridging laws
can show good agreements with the results of thetension test in which the specimens were fabricated byhorizontal and vertical casting. However, it is difficult
to use calculated bridging laws directly for evaluationof characteristics of FRCC elements such as beams
and columns, because the shape of bridging laws
which are expressed by tensile stress and crack widthis strongly affected by fiber orientation. It isconsidered that simpler models for bridging laws
make evaluations of FRCC elements easier.
Many types of tensile stress–crack width models forbridging laws can be considered. For example,

652

Modeling of Bridging Law for PVA Fiber-Reinforced Cementitious
Composite Considering Fiber Orientation

multi-function models, multi-linear models, tri-linear
models, and bi-linear models have been introduced fortensile stress–strain models of FRCC [2]. In this study,a tri-linear model shown in Fig. 1 is chosen todescribe the characteristic points in calculated
bridging laws considering the phenomena those occur
in single fiber pullout properties. The modeled
bridging laws are used for section analysis of bending
test specimens to verify their adaptability. The
modeled bridging laws are characterized by fiber
orientation, so the bending specimens are fabricatedby three casting methods to vary the fiber orientation.
At first, calculation method of bridging laws
proposed by authors is briefly introduced in next
chapter.

2. Calculation Method of Bridging Law

Tensile stress

The authors have proposed the bridging law, whichis expressed by tensile stress–crack width relationship,considering the influence of fiber orientation [3]. The
bridging law is calculated by the summation of thepullout properties of each single fiber in crack surface.
The model of the pullout load–pullout displacement
relationship of the single fiber is shown in Fig. 2. The
pullout load–pullout displacement relationship ismodeled by tri-linear model considering chemical andfrictional bond between PVA fiber and matrix [5]. At
the time of the first peak load Pa, the chemical bond isdebonded over the entire length of fiber. After that,the pullout load increases to maximum load Pmax
because of the frictional bond. The pullout load

becomes zero when the crack width corresponds to theembedded length of the single fiber, lb. The crack
width at the first peak load, wa, is twice the value ofthe pullout displacement, pull. The crack width at the
maximum load, wmax, is 1.5 times the pullout
displacement because the pullout displacement on theend of long embedded length starts decreasing when
the pullout load begins to decrease on the end of shortembedded length.
The pullout property of single fiber varies due tosnubbing effect and fiber strength degradation byorientation angle, . Snubbing effect shows theincrement of pullout load due to the reaction force atthe embedding edge of fiber when fiber hasorientation angle [4]. Fiber strength degradationshows the decreasing of fiber strength in the case ofpolymer fiber due to the surface of fiber roughed by
the embedding edge when fiber is embedded
obliquely with normal direction of crack surface [5].
As shown in Fig. 2, though the pullout load increases
with increasing of orientation angle by the snubbing
effect, fiber tends to rupture by the fiber strength
degradation if orientation angle becomes large.
In order to give the orientation angle to each single
fiber at crack surface in the calculation of the bridging
law, a PDF (probability density function) has been
proposed by Kanakubo et al. [3]. The PDF expresses
the fiber orientation distribution using elliptic
distribution. The fiber orientation varies by the valueof orientation intensity k (ratio of the two radiiof elliptic function) and principal angle r (argument of

 max
2
 max  2

Crack widthFig. 1

Proposed tri-linear model for bridging law.

 tu

Pullout load

Modeling of Bridging Law for PVA Fiber-Reinforced Cementitious
Composite Considering Fiber Orientation

=0

Rupturestrength

Pmax
Pa
lb

 :medium

Pullout load

FiberMatrix

 pullwa

wa wmax

Pullout load

At Pa
lb

653

=2  pullShort
embedded side

At Pmax

 pullwmax=1.5 pull

 :large
RuptureAt slipping out
Fiber

Crack widthlb

MatrixFig. 2 Pullout model of a single fiber.

one radius of elliptic function). The randomorientation is given by k = 1. When the value of k islarger than 1, fibers tend to orient toward r. Whenthe value of k is smaller than 1, fibers tend to orient
toward the perpendicular to r.
Tensile stress is calculated by the summation of thepullout load of each single fiber at crack surface asgiven in Eqs. (1) and (2).

 bridge PbridgeAm
Vf
Af  Pij ( w, )  pxy (i )  pzx ( j )
h

j

(1)

i

where,

bridge = tensile stress;
Pbridge = bridging force (= total of pullout load);
Am = cross-sectional area of matrix;
Vf = fiber volume fraction;

 = fiber angle to x-axis;
 = angle between x-axis and projected line of thefiber to x-y plane; px ( yh , zh )      (y  z )P  Ppull  e f   Prup  e  f '

Af = cross-sectional area of a single fiber;
P = pullout load of a single fiber;
Ppull = pullout load of a single fiber at a zero fiber
angle;
Prup = pullout load of a single fiber at rupture at azero fiber angle;
F = snubbing coefficient;f’ = fiber strength reduction factor;
pxy, pzx = probability by elliptic distribution;px = probability of fiber distribution along x-axis;

 = angle between x-axis and projected line of the(2)

fiber to z-x plane.
The PDF, px (y, z), gives the probability for the
existence of the fiber in the x-axis direction. In thisstudy, px (y, z) is assumed to be constant. This meansthat the fibers are randomly distributed along thelongitudinal direction of the specimen.
The calculated bridging laws for the orientation

654

Modeling of Bridging Law for PVA Fiber-Reinforced Cementitious
Composite Considering Fiber Orientation

intensity k from 0.1 to 10 are shown in Fig. 3. The
parameters for the calculation are listed in Table 1.
The principal angle r is set to 0˚ (axial direction of
the specimen). The curves in Fig. 3 do not include thecrack strength to exhibit the stress due to only
bridging force of fibers. As shown in Fig. 3, tensilestress significantly drops after the maximum stress,
then moderate decrement follows. The significant
drop and moderate decrement of the tensile stress inthe calculation after the maximum stress is caused byrupture and pullout of fibers, respectively. The
maximum tensile stress in the bridging law
remarkably increases with the increment of the valueof k, i.e., stronger fiber orientation to the normal
direction of crack surface.

3. Modeling of Bridging Law
As described in the former chapter, the calculated
bridging laws are strongly affected by fiber orientation.
Proposing simpler models for bridging laws can makeevaluations of FRCC elements easier. In this study, a

tri-linear model shown in Fig. 1 is newly proposed toexpress tensile stress–crack width relationship of
FRCC after first cracking considering the fiber
orientation.
From Fig. 3, the bridging law is characterized bythree parts, i.e., the part from the origin to maximum
tensile stress, the part of the significant drop of tensilestress after maximum tensile stress, and the part ofmoderate decrement of tensile stress. Therefore, thetri-linear model of the bridging law is considered to besuitable as shown in Fig. 1. The model has fiveparameters: the maximum tensile stress, max, thecrack width at max, max, the second point tensilestress after the significant drop of stress, 2, the crack
width at 2, 2, and the crack width at the loss ofstress, tu. tu is constant value, because fiber iscompletely pulled out when crack width becomes halfof fiber length.
Remaining four parameters in the model are
expressed as a function of the orientation intensity
k to simplify the modeling of the bridging law. The

Tensile stress (MPa)

4
3
2
1
0

Fig. 3
Table 10

1
Crack width (mm)Calculation result of bridging law.
Parameters for calculation of bridging law.

Parameter
Fiber volume fraction, Vf
Fiber length, lf
Fiber diameter, df
First peak load, Pa
Crack width at Pa, wa
Maximum load, Pmax
Crack width at Pmax, wmax
Fiber effective strength
Snubbing coefficient, fFiber strength reduction factor, f’

Input value
2.0%
12 mm
0.10 mm
1.5 N
0.20 mm
3.0 N
0.45 mm
569 MPa
0.5
0.3

2

k=10
k=1
k=0.1

Modeling of Bridging Law for PVA Fiber-Reinforced Cementitious
Composite Considering Fiber Orientation

relationship between the parameters for stress and k
are shown in Fig. 4. The relationship between theparameters for crack width and k are shown in Fig. 5.
The dotted lines in all figures exhibit the regression
calculation results for the values of the parameters by
the least squares method. The solid lines exhibit themodified regression calculation result to simplify the
relational expression between each parameter and k,
and they are shown in each figure. The stress values inFig. 4 are decided to be expressed by the function
which starts from the origin because the bridging
force becomes zero in the case of k → 0, i.e., nobridging fiber exists. The function for max is decided
to pass the point of calculated value of max at k = 1(random orientation). From the calculation results ofbridging law, 2 corresponds with the crack width at
the maximum pullout load of single fiber in Table 1,
i.e., constant value of 0.45 mm.

4. Bending Test of FRCC Varying Fiber
Orientation
The bending test of PVA-FRCC is conducted toverify the adaptability of the modeled bridging law

through section analysis. The bending test specimens
are fabricated by three casting methods to vary thefiber orientation.
4.1 Specimens and Used Materials
The dimension of the specimens applied in thisstudy is shown in Fig. 6. Specimens are the notchedbeams which have the cross-section of 100 mm × 100
mm and a notch with a depth of 30 mm and a width of
5 mm by a concrete cutter after FRCC become
hardened. The position of a notch and a casting
direction is as shown in Fig. 6. These dimensions andmanufacturing procedures follow ISO 19044 [6]. The
experimental parameter is the placing method of
FRCC using the compacting vibrator explained after.
Six specimens were manufactured for each parameter,
and total of 18 specimens were tested.
PVA fiber of 0.10 mm diameter and 12 mm length
were utilized in this study. The mix proportion of
PVA-FRCC is shown in Table 2. The volume fractionof PVA fiber is 2.0%. FRCC in this study has
self-compacting behavior. The average of compressive
strength in the material age at bending tests was 34.6 MPa

max (MPa)

4
3

ma x

2
1
0

0

y=2.0x0.30
510Orientation intensity k

42 (MPa)

3

2

2
1
0.73

0

Fig. 4Stress parameters for tri-linear model.

0655

y=0.60x
510Orientation Intensity k

656

Modeling of Bridging Law for PVA Fiber-Reinforced Cementitious
Composite Considering Fiber Orientation
0.5

max (mm)0.4

 ma x

0.3
0.2
0.1
0

0

y=0.20x 0.18
510Orientation intensity k

0.5

2 (mm)

0.4
0.3
0.2
0.1
0

2
0

5
Orientation intensity k

Fig. 5

Crack width parameters for tri-linear model.

Fig. 6

Notched beam specimen for bending test (unit: mm).

Table 210

Mix proportion of PVA-FRCC.

Fiber volume fraction Water by binder ratio

Sand by binder ratio

Water
2.0%
0.39
0.50
380
Cement: High early strength Portland cement, Fly ash: Type II of JIS A 6202.
Sand: Size under 0.2 mm, Super plasticizer: Binder × 0.6%.

by 100 mm × 200 mm cylinder test pieces.
4.2 Placing of FRCC
Many researches have studied the effects of fiber
orientation on the mechanical characteristics of FRCC,

Unit weight (kg/m3)
CementFly ash
678
291

Sand
484

including FRC (fiber-reinforced concrete). The
scheme of the current approach to evaluate the fiber
orientation has considered the casting method,
fresh-state properties, flow, vibration, and formwork
geometry (e.g. [7]). The authors have studied the657

Modeling of Bridging Law for PVA Fiber-Reinforced Cementitious
Composite Considering Fiber Orientation

effect of a compacting vibrator to bending
characteristics of FRCC beams [8]. The bending
capacity and ductility increases by applying acompacting vibrator after pouring FRCC. It issuggested that the fiber orientation tends to differ ineach specimen by observing the specimen sections
after loading. So, the method of applying acompacting vibrator is also introduced in this study tovary the fiber orientation of specimens.
Fig. 7 shows the three series of compacting
methods of FRCC: (a) SC (self-compacting) without
vibrating; (b) VF (vibrator-fix); and (c) VM(vibrator-move). A vibrator with a 24.5 mm diameterrod, and vibrating frequency of 200 Hz, which iscommonly used for compacting of conventional
concrete, was used.
SC is standard placing method in which FRCC iscontinuously poured from the edge of the mold withthe slope of 1/33. In VF compacting, the vibrator is set
with vibration at the center of the specimen after
FRCC is poured into the mold. The vibration period is
10 s. VM compacting is the method in which theyxVibrator

vibrator is moved and reciprocated with vibrationfrom the end of the mold to the other end after FRCC
pouring. The vibration period is also 10 s.
The examples of photos of the specimens of VF andVM are shown in Fig. 8. As shown in Fig. 8, in thecase of VF, FRCC matrix showed a circular motion
with a central focus on the vibrating rod, i.e., fibers
tend to orient concentrically. In the case of VM,
FRCC matrix flowed longitudinally following
movement of the vibrating rod. These observations
indicate that fibers in FRCC tend to orient toward theaxial direction along the flow.
4.3 Method of Loading and Measurements
Fig. 9 shows the bending test setup. Three-point
bending tests were conducted based on ISO 19044 [6]using the displacement controlled universal loadingmachine of 2,000 kN capacity. The speed of thecross-head was set to 0.5 mm/min. Measurement
items were load, axial deformation in central part ofthe specimen (gauge length = 100 mm) using two
pi-type displacement transducers, and the LPD (loading
Fixed

Vibrator

Reciprocating

z
Mold

FRCC

FRCC

Mold
100

Notch
(Cutting after hardening)

200

Fig. 7(a) SC (self compacting)Compacting method (unit: mm).

(b) VF (vibrator-fix)

Fig. 8(a) VF (vibrator-fix)
Manufacturing and vibrating of specimen.

400(c) VM (vibrator-move)(b) VM (vibrator-move)

100

658

Fig. 9Modeling of Bridging Law for PVA Fiber-Reinforced Cementitious
Composite Considering Fiber Orientation

Bending test setup.

point displacement) using a LVDT as shown in Fig. 6.
4.4 Test Results
The examples of the specimens of each placing
method after loading are shown in Fig. 10. All
specimens had plural cracks and fractured withlocalizing the opening of one crack after maximum
load. In the case of SC specimens (self-compacting),
rectilinear cracks took place perpendicularly to axial
direction of the specimen. In the VF specimens(vibrator-fix), curved cracks were observed. It isconsidered that the curved cracks occurred due to thefiber orientation like concentric circles centering thepoint of vibrating. This assumes that cracks tend toprogress along the weaker parts in the matrix, in
which fibers orient toward crack direction and weakerbridging effect provides. In the case of VM specimens(vibrator-move), there were more cracks than those in
the specimens SC and VF. It is suggested that thetendency of fiber orientation toward the axial directionin the VM specimen causes more cracks by stronger
bridging effect.
Fig. 11 shows the load–CMOD (crack mouth
opening displacement) curves of all specimens.
CMOD was calculated as the axial deformation at theundersurface of the specimen from the measured axial
deformation by pi-type displacement transducers. The
points of the maximum load are plotted by circles. As
shown in Fig. 11, load gradually increased after first
cracking in all specimens. After the peak load, theload slightly decreased with the repetition of increase
and decrease of the load. Table 3 shows the summaryof the bending test results. Fracture energy in Table 3
is calculated by the following Eq. (3).
W
(3)
GF 
Alig
where,
GF = fracture energy;
W = area below load–LPD curve up to 15 mm;
Alig = area of ligament.
The maximum loads of the VM specimens aretwice larger than those of the SC specimens, and thefracture energies of the VM specimens are largest ofall. It is considered that the bridging effect isimproved by moving a vibrator. In the case of the VF
specimens, the maximum loads show larger scattering
comparatively.

5. Adaptability of Modeled Bridging Laws
The modeled bridging laws are used in section
analysis to verify their adaptability through theevaluation of bending strength of the specimens
explained in former chapter.
5.1 Section Analysis
In order to evaluate the bending strength, thesection analysis based on the modeled bridging law isconducted. Fig. 12 shows whole stress–crack widthmodel applied for the section analysis. The tensilestress–crack width model is the tri-linear model
proposed in this study expressed as the functions
shown in Fig. 4 and Fig. 5. The compression side isassumed to keep elasticity. The section analysis iscarried out based on the assumption that plain sectionModeling of Bridging Law for PVA Fiber-Reinforced Cementitious
Composite Considering Fiber Orientation

659(a) SC (self compacting)Fig. 10(b) VF (vibrator-fix)
Examples of specimens after loading.

(c) VM (vibrator-move)SC(self compacting)VF(vibrator-fix)
VM(vibrator-move)
Max. load

10

Load P (kN)

8
6
4
2
0
Fig. 110

5
10CMOD (mm)15

Load–CMOD curves.

remains plain under considering the plain section’s
deformation. Firstly, arbitrary angle of rotation is
given. Secondly, the crack width of each element incross-section is calculated from linear distribution of
deformation and the stress in each element ofcross-section is obtained from Fig. 12. Finally, neutral
axis satisfying equilibrium condition is foundnumerically and bending moment is calculated. The
section analysis is conducted by varying orientation
intensity k in the bridging law model.
The compressive stress–compressive deformation
model is elastic based on the compression test resultsof cylinder test pieces. However, compressive strain

instead of deformation is obtained from the compression
test. In order to convert the strain into the deformation,equivalent compressive length lc,eq is introduced. The
equivalent compressive length is determined to fit theinitial slope between the result of section analysis andmeasured load–CMOD curves in the bending test. The
stiffnessincompressive
stress–compressive
deformation model is given by elastic modulus
measured in compression test (Ec) divided by lc,eq.
5.2 Comparison of Analysis and Test Results
The comparison of the maximum bending moment
between the results of section analysis and bending

660

Table 3
Placing
methodSCVF

VM

Modeling of Bridging Law for PVA Fiber-Reinforced Cementitious
Composite Considering Fiber Orientation
Bending test results.
ID

At maximum load
CMOD
(mm)
2.19
2.24
2.58
2.65
2.11
2.35
0.24
1.84
1.61
1.66
2.71
1.93
2.01
1.96
0.40
1.40
3.15
2.72
3.71
2.55
3.45
2.83
0.82

Max. load
(kN)
4.53
4.14
3.99
4.14
4.50
4.70
4.33
0.28
6.20
6.46
5.95
6.89
5.57
4.07
5.85
0.98
9.05
9.05
9.08
8.72
9.41
8.88
9.03
0.23

SC-1
SC-2
SC-3
SC-4
SC-5
SC-6
Average
STDV
VF-1
VF-2
VF-3
VF-4
VF-5
VF-6
Average
STDV
VM-1
VM-2
VM-3
VM-4
VM-5
VM-6
Average
STDV(MPa) Tension

max
2

Tri-linear modeltu (mm)

Ec / lc,eq max 2Compression
Elastic modelStress–crack width (deformation) model for section analysis.
Maximum moment (kN・m)

Fig. 12

0.8
0.6
0.4
0.2
0

Analysis result
Test result
N
SC

VF
VF

VM
VM

Fig. 13 Maximum bending moment by section analysis and bending test results.

k=6
k=5
k=4
k=3
k=2
k=1
k=0.8
k=0.6
k=0.4
k=0.2

Fracture energy(N/mm)
4.15
3.58
3.47
3.17
3.26
3.51
3.52
0.34
4.46
4.21
4.46
7.65
4.24
3.02
4.67
1.55
10.2
9.01
8.20
8.69
9.74
7.03
8.81
1.13

Modeling of Bridging Law for PVA Fiber-Reinforced Cementitious
Composite Considering Fiber Orientation

test is shown in Fig. 13. The dotted lines indicate themaximum bending moment calculated by section
analysis using each orientation intensity k from 0.2 to6. The average maximum bending moments in thebending test are plotted by squares with error bars
(standard deviation). As shown in Fig. 13, the analysis
results in the case of k = 0.4, k = 1, and k = 5 show
good agreements with the test results of the SC
specimens, VF specimens, and VM specimens,
respectively. The difference of bending strength dueto the fiber orientation is expressed by the section
analysis using the bridging law considering fiber
orientation.

6. Conclusions
The authors have proposed the method of
calculation of bridging law, which is expressed bytensile stress–crack width relationship, considering theinfluence of fiber orientation. The calculated bridging
law is strongly affected by fiber orientation, which isfeatured by orientation intensity k. A tri-linear modelis newly proposed to express the tensile stress–crack
width relationship considering the fiber orientation.
Through the bending test, in which the specimens
were fabricated by three casting methods to vary thefiber orientation, the followings are found out:
(1) The parameters that give the characteristic
points of the tri-linear model are proposed asfunctions of orientation intensity k.
(2) The maximum loads of the VM specimens (acompacting vibrator is reciprocated longitudinallyafter FRCC pouring) are twice larger than thoseobtained by SC specimens.
(3) The results of section analysis, which is
conducted to verify the adaptability of the proposed

661

model, can present the difference of bending strength
due to the fiber orientation.

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

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