1

FIBER REINFORCED CONCRETE

N. Banthia

Synopsis: The usefulness of fiber reinforced concrete (FRC) in

various civil engineering applications is indisputable. Fiber

reinforced concrete has so far been successfully used in slabs on

grade, shotcrete, architectural panels, precast products, offshore

structures, structures in seismic regions, thin and thick repairs,

crash barriers, footings, hydraulic structures and many other

applications. This paper presents a brief state-of-the-art report on

mechanical properties and durability of fiber reinforced concrete.

In particular, issues related to fiber-matrix interaction,

reinforcement mechanisms, standardized testing, resistance to

dynamic loads, and transport properties are discussed.

Keywords: FRC, fiber reinforced concrete, fiber-matrix

interaction, dynamic loads

2

Nemkumar Banthia is a Professor and Distinguished University

Scholar at the University of British Columbia, Vancouver, Canada. A

fellow of the ACI, the Canadian Soc. of Civil Engg., Indian Concrete

Institute and the Canadian Academy of Engineering, his awards include

the Wason Medal of the ACI and the Solutions through Research Award

of the Innovation Council of British Columbia.

INTRODUCTION

Compared to other building materials such as metals and polymers,

concrete is significantly more brittle and exhibits a poor tensile

strength. Based on fracture toughness values, steel is at least 100

times more resistant to crack growth than concrete. Concrete in

service thus cracks easily, and this cracking creates easy access

routes for deleterious agents resulting in early saturation, freeze-

thaw damage, scaling, discoloration and steel corrosion.

The concerns with the inferior fracture toughness of concrete are

alleviated to a large extent by reinforcing it with fibers of various

materials. The resulting material with a random distribution of

short, discontinuous fibers is termed fiber reinforced concrete

(FRC) and is slowly becoming a well accepted mainstream

construction material. Significant progress has been made in the

last thirty years towards understanding the short and long-term

performances of fiber reinforced cementitious materials, and this

has resulted in a number of novel and innovative applications.

There are currently 200,000 metric tons of fibers used for concrete

reinforcement. Table 1 shows the existing commercial fibers and

their properties. Steel fiber remains the most used fiber of all (50% of

total tonnage used) followed by polypropylene (20%), glass (5%)

and other fibers (25%).

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PERFORMANCE CHARACTERISTICS OF FIBER

REINFORCED CONCRETE

Reinforcement Mechanisms

Concrete carries flaws and micro-cracks both in the material and at

the interfaces even before an external load is applied. These defects

and micro-cracks emanate from excess water, bleeding, plastic

settlement, thermal and shrinkage strains and stress concentrations

imposed by external restraints. Under an applied load, distributed

micro-cracks propagate coalesce and align themselves to produce

macro-cracks. When loads are further increased, conditions of

critical crack growth are attained at the tips of the macro-cracks and

unstable and catastrophic failure is precipitated.

The micro and macro-fracturing processes described above, can be

favorably modified by adding short, randomly distributed fibers of

various suitable materials. Fibers not only suppress the formation of

cracks, but also abate their propagation and growth.

Soon after placement, evaporation of the mix water and the

autogenous process of concrete hydration create shrinkage strains in

concrete. If restrained, this contraction can cause stresses far in

excess of those needed to cause cracking. In spite of every effort,

plastic shrinkage cracking remains a serious concern, particularly in

large surface area placements like slabs on grade, thin surface repairs,

patching and shotcrete linings. With large surface areas, fibers

engage water in the mix and reduce bleeding and segregation. The

result is that there is less water available for evaporation and less

overall free shrinkage [1]. When combined with post-crack bridging

capability of fibers, fibers reduce crack widths and cracks areas when

concrete is retrained [2], (Fig. 1).

In the hardened state, when fibers are properly bonded, they interact

with the matrix at the level of micro-cracks and effectively bridge

these cracks thereby providing stress transfer media that delays their

coalescence and unstable growth (Fig. 2). If the fiber volume fraction

is sufficiently high, this may result in an increase in the tensile

4

strength of the matrix. Indeed, for some high volume fraction fiber

composite [3], a notable increase in the tensile/flexural strength over

and above the plain matrix has been reported (Fig. 3). Once the

tensile capacity of the composite is reached, and coalescence and

conversion of micro-cracks to macro-cracks has occurred, fibers,

depending on their length and bonding characteristics continue to

restrain crack opening and crack growth by effectively bridging

across macro-cracks. This post-peak macro-crack bridging is the

primary reinforcement mechanism in the majority of commercial

fiber reinforced concrete composites.

Based on the discussion above, it emerges that fiber-reinforced

cementitious composites can be classified into two broad categories:

normal performance (or conventional) fiber-reinforced cementitious

composites and high-performance fiber-reinforced cementitious

composites. In FRCs with low to medium volume fraction of fibers,

fibers do not enhance the tensile/flexural strength of the composite

and benefits of fiber reinforcement are limited to energy absorption

or ‗toughness‘ enhancement in the post-cracking regime only. For

high performance fiber reinforced composites, on the other hand,

with a high fiber dosage, benefits of fiber reinforcement are noted in

an increased tensile strength, strain-hardening response before

localization and enhanced ‗toughness‘ beyond crack localization.

Fiber-Matrix Bond

As in any fiber reinforced composite, fiber-matrix bond in FRC is of

critical importance. However, unlike fiber reinforced polymers

(FRPs) used in aerospace and automobile industries where fibers are

employed to enhance strength and elastic modulus, in FRCs,

‗toughness‘ or energy absorption capability is of primary interest.

Therefore, inelastic bond failure mechanisms such as interfacial

crack growth, crack tortuousity and fiber slip are of greater relevance.

Fiber pull-out tests are often performed to assess fiber efficiency in

FRC and in such tests fiber bond and slip are monitored

simultaneously. Fig. 4 shows such a test, and the bond-slip curves

obtained [4].

5

For a fiber embedded in a cementitious matrix and subjected to a

pull-out load (Fig. 5), shear-lag will occur and interfacial debonding

will commence at the point of fiber entry which will slowly

propagate towards the free end of the fiber. Thus, some energy

absorption will occur at the fiber-matrix interface while the bond is

being mobilized and the fiber prepares to slip. Early in the

development of fiber reinforced concrete it became apparent that for

large, macro-fibers with small surface areas, a straight fiber will pull-

out at low values of interfacial stress and will generate stress in fiber

far below its tensile strength. Most commercial macro-fibers of steel

and other materials (polypropylene, for example) are now deformed

to enhance their bond with the surrounding matrix. However, even

here there is a limit. If deformed excessively, fibers may develop

stresses that exceed their strength and fracture in the process (Fig. 3).

The energy absorption in such cases is limited, and although some

fiber slippage may precede fracture, poor toughening ensues. For

maximized fiber efficiency, a pull-out mode of fiber failure where

pull-out occurs at a fiber stress close to its tensile strength is

preferred. It is important to mention that fiber failure mode is highly

dependent on the angle at which fiber is inclined with respect to the

direction of the pull-out force.

Fundamental Fracture Studies and Modeling

In the case of classically brittle materials like glass, Linear Elastic

Fracture Mechanics applies and fracture can be completely defined

by a single parameter called the ‗critical stress intensity factor‘, KC.

In micro-fracturing, strain-softening material like concrete, one

parameter description of fracture is not possible and multi-

parameter fracture criterion have been proposed [5,6]. In the case

of fiber reinforced concrete, in addition to crack closing pressure

due to aggregate interlocking, fiber bridging occurs behind the tip

of a propagating crack where fibers undergo bond-slip processes

and provide additional closing pressures. The fracture processes in

fiber reinforced cement composites are therefore even more

complex and advanced models are needed to simulate these

processes. Attempts have been made to model fracture in FRC

using the cohesive crack model [7] as well as the J-integral [8].

However, strictly speaking, these are only crack initiation criteria

6

and fail to define conditions for continued crack growth. To define

both crack initiation and growth, there is now general agreement

that a continuous curve of fracture conditions at the crack tip is

needed as done in an R-curve [9]. An R-curve (Figure 6) is a

significantly more suitable representation of fracture in FRCs, as

one can monitor variations in the stress intensity as the crack

grows and derive a multi-parameter fracture criterion.

A contoured double cantilever specimen is often used to obtain R-

Curves for FRC. A typical test is shown in Fig. 7, and the resulting

R-Curves are shown in Fig. 8 [10].

Standardized Tests for Toughness Measurements

Characterization of toughness (or energy absorption capability) of

FRC through standardized testing remains a hotly debated topic.

There is still no general agreement on how the toughness of fiber-

reinforced concrete should be measured [11-13]. ASTM has two

standards [14,15] and Japan Society for Civil Engineering (JSCE)

Standard SF-4 [16] is also often used. These three techniques and

their analysis schemes are compared in Table 2. Unfortunately, they

all treat toughness differently and there is little cross relationship

between the toughness parameters they produce [17].

Blast and Impact Resistance

Since 9/11, there has been an increased interest in developing

materials with enhanced resistance to explosive and impact loads.

Testing has clearly demonstrated that the ideal way to enhance the

impact resistance of concrete is by fiber reinforcement. Fibers

enhance the post-fracture stress transfer capability in concrete,

enhance dynamic fracture toughness, decrease dynamic crack

velocities and increase the absorption of energy under impact loads.

Drop Weight Impact Tests [18] are generally performed to measure

the resistance of fiber reinforced concrete to impact loads. In most

modern impact test systems, sufficient instrumentation is provided

such that loads, deformations, and velocities are simultaneously

measured; these are needed for a later analysis of the data. One

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major issue that needs to be dealt with is that of inertial loading.

Specimen accelerates during a test and all measurements are made

while the specimen is still under acceleration. One commonly

adopted technique is to carry out direct measurements of

accelerations, and then use the principle of virtual work to derive

expressions for generalized inertial loads. For a simply supported

plate specimen impacted in the center, the generalized inertial load

(Pi(t)) is given by [19]:

l

.xt).cosecl/2,(x,u

4

hl(t)P

2

i

(1)

where, l is width (also length) of the plate, is the mass density, h

is the thickness of the plate and ü(x,y,t) = acceleration at any point

(x,y) on the plate at time t. Once the generalized inertial load is

obtained, the plate can be modelled as a Single Degree of Freedom

(SDOF) system and the generalized bending load can be obtained

from the Equation of dynamic equilibrium,

P (t) P (t) P (t)b t i (2)

Similar expressions have been developed for beam specimens [18]

and other geometries [20].

For fiber reinforced concrete, while an improvement in impact

properties is widely reported, on a worrisome note, steel fibers are

reported [21] to fracture across cracks at high rates of loading and

thus produce a brittle response at very high strain-rates. As seen in

Fig. 9 and 10, SFRC may show increased brittleness under very

high strain rates.

The exact reasons of the observed brittleness of some FRC

materials under impact can be understood only via fundamental

testing of bond-slip mechanisms, fracture studies and modeling

[22]. In a recent study [23], a model proposed by Armelin and

Banthia [24] was adopted to predict the load-displacement

response of beam under impact (Fig. 11). The compressive strain,

o, at the top-most fiber of the specimen leads to an axial

shortening, o, as shown. This in turn leads to stress, c, in the

8

uncracked concrete. On the other hand, it results in fiber slippage,

wi, below the neutral axis and corresponding forces, fi, as the fibers

pull-out. Thus, the flexural load carried during the post-crack

phase is obtained by satisfying the equilibrium of moments:

l

MP e2 (3)

The equilibrating moment, Me, may be calculated by summing the

moments generated by concrete stresses and the individual

moments generated by the N individual fibers bridging the crack

below the neutral axis. It follows from Fig. 11, that

'

0 1

0.

c N

ic fdyb (equilibrating forces) (4)

N

ii

c

ce yfydybM10

.

'

(equilibrating moments) (5)

where b is the width of the beam, c’ is the depth of the uncracked

section and y is the distance from the neutral axis.

In the model, the pull-out force in each fiber (fi) is expressed as a

function of the crack width, wi, according to the average pull-out

force versus slip (or crack width) relationship obtained

experimentally at the full embedment length, le=l/2. To enable this,

single fibers must be pulled out from a concrete matrix at various

inclinations with respect to the pull-out load. The bond-slip

response is then represented using the Ramberg-Osgood

formulation so that the force carried by each fiber may be

expressed in terms of its orientation, i and the slip, wi, as follows:

CC

i

ipiii

Bw

AAwEwf

1

1

1, (6)

where the constants A, B, C and Ep, are obtained for each

orientation through the Ramberg-Osgood formulation.

Recognizing that the average force in the fibers at a layer which is

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at a distance ‗y‘ from the neutral axis is averaged over the entire

range of embedment and inclination that is possible, the value of

‗fi‘ in Equations 4 and 5 may be computed as follows (24):

wf

wfwfwfwf

wff geometryi

4

1

222

1 90

5.67455.22

0 (7)

or

wf

wfwfwfwfwfwf

wff geometryi

6

1

222

1 90

7560453015

0 (8)

The resulting model prediction is plotted together with the

experimental flexural response under impact in Fig. 12. Not

surprisingly, the analytical response is monotonic and predictions

are not excellent. Nevertheless the model can approximate the

experimental response, and predict both peak load and toughness.

For proper predictions, one needs to somehow engage information

on crack velocities and changes in the stress-intensity factors at the

tip of a fast moving crack. Such attempts are currently underway.

Bio-Inspired FRCs for Longer Service Life

The decreasing life span of concrete structures is becoming an

issue of greater and greater importance for societies. The primary

problem is the corrosion of the steel in reinforced concrete

structures. Chloride penetration and carbonation are the primary

reasons for such corrosion and any measures aimed at mitigating

the ingress of chlorides or CO2 into the body of concrete are

expected to significantly enhance the durability of concrete

structures. These deleterious agents enter the body of concrete

through one of the three transport mechanisms: diffusion, capillary

sorption and permeability—of these, the permeability is considered

as the dominant mechanism. Any measures adopted to reduce

permeability of concrete will therefore help in preserving

durability. Results have indicated that permeability, in turn, is

highly dependent upon cracking in concrete and an increase in the

crack width will not only produce a highly permeable concrete (Fig.

13) but also enhance the possibility of rebar corrosion, Figure 14,

Bentur [25].

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There is an increased interest these days in bio-inspired fiber

reinforced cementitious composites. These are based on bio-

degradable and recyclable natural fibers (such as cellulose) and are

both environmentally friendly and sustainable. Current research at

The University of British Columbia is focussing on such

composites, and the preliminary indications are that such bio-

inspired materials are highly promising in building and

regenerating a sustainable infrastructure.

The influence of an externally applied stress on the permeability of

concrete remains poorly understood. Banthia and co-workers [26]

& [27], developed a novel technique of measuring the permeability

of concrete under an applied stress and investigated the benefits of

fiber reinforcement. The permeability cell was mounted directly in

a 200 kN hydraulic Universal Testing Machine (UTM) such that a

uniform compressive stress could be applied directly on the

concrete specimen housed in the cell. The water collected was

related to the coefficient of water permeability (Kw) by applying

Darcy‘s law:

hA

QLKw

(9)

Kw = Coefficient of water permeability (m/s), Q = Rate of Water

Flow (m3/s), L = Thickness of specimen wall (m), A = Permeation

area (m2) and h = Pressure head (m).

Their data are plotted in Fig. 15. Notice that under conditions of

no-stress, fibers reduce the permeability of concrete, and the

reduction appears to be proportional to the fiber volume fraction.

Data further indicates that stress has a significant influence on the

permeability of concrete. When stress was first increased to 0.3fu,

both plain and FRC showed a decrease in the permeability.

However, when the stress was increased to 0.5fu, plain and FRC

showed very different trends. At 0.5fu, the permeability of plain

concrete increased substantially over that of the unstressed

specimen, but for FRC, while there was an increase in the

permeability over 0.3fu, the permeability still stayed below that of

the unstressed specimen.

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The above observations can be related to cracking. At 0.3fu, it is

conceivable that in both plain and FRC, there is no discernible

cracking that can affect the flow of water. However, at 0.3fu, the

stress-strain response for both plain and FRC would become non-

linear indicating the presence of cracking. As given by the

Poiseuille Law, Edvardsen [28], the flow of water through cracks

is proportional to the cube of the crack width. In the case of FRC,

one can expect the fibers to suppress cracking and hence maintain

the rate of flow similar to an unstressed specimen. When combined

with the phenomenon of ‗pore compression‘, this implies that the

permeability of FRC under stress can in fact be lower than that of

an unstressed specimen.

Bhargava and Banthia [27] extended the permeability data

described above towards service life prediction. Most service life

prediction models for concrete involve the use of diffusion

coefficients Tutti [29]. Unfortunately, studies relating different

transport coefficients are rare. In particular, experimental data

relating permeability and diffusion coefficient is lacking, and only

a theoretical correlation can be established between these two

coefficients via a correlation constant, as follows:

Empirical equations for the permeability coefficient were proposed

by Hedegaard et al. [30] and for diffusion coefficient were

proposed by Hansen et al. [31] as follows:

0.4

w

f31.0c3.4expKw (10)

0.7

3.0

107.1 w

fc

xD (11)

where,

Kw= water permeability coefficient (m/s)

D = Chloride ion diffusion coefficient, in cm2/s

c = cement content of concrete, in kg/m3

12

w= water content of concrete, in kg/m3

f = fly-ash content of concrete, in kg/m3

By substituting the values of c, w and f for the concrete mixture

used in the permeability tests in Eqs. (10) and (11), one obtains

Kw=1.07x10-10

(m/s) and D = 7.89x10-13

(m2/s).

Further, the permeability K (m2) of a single straight pore with

effective pore radius effr embedded in a medium of cross-sectional

area A can be related to effective pore radius by assuming Hagen-

Poiseuille’s law to be valid for small pores.

A

rK

eff

8

4 (12)

where effr is the effective pore radius defined as the radius of the

effective pores which take part in the transport. Also, the diffusion

coefficient can be related to the area fraction of effective pores as,

A

rDaDD

eff

oeffo

2 (13)

where effa = is the area fraction of effective pores

oD = is the diffusion coefficient in a bulk fluid

Assuming that the effective pore radius in Eqs. (12) and (13) is the

same, a general relationship between permeability K (m2) and

diffusion coefficients D (m2/s) emerges,

DD

rK

o

eff

8

2

(14)

Further, it is to be noted that an interconnected pore system is

necessary for a continuous network of flow paths to be available

for various transporting media. In saturated conditions, the steady

state flow coefficient can be related to the water permeability

coefficient as the two processes occur simultaneously,

13

g

KK w

(15)

Using Eqs. (14) and (15), the water permeability coefficient wK

(m/s) and the diffusion coefficient D (m2/s) can be related as,

DD

grK

o

effw

8

2

(16)

Where Kw as before is the water permeability coefficient (m/s),

D is the diffusion coefficient (m2/s),

reff is the effective pore radius,

is the viscosity of water (Ns/m2),

is the density of water (kg/m3) and,

g is the gravity (m/s2)

This equation corresponds to Katz-Thompson Equation, Garboczi

[32], and is based on the assumption that the effective radius

affecting the permeability and the diffusion coefficient is the same.

Equation (16) can be further modified to consider the effect of

stress and the fibers on concrete. Since the permeability coefficient

is proportional to the fourth power of effective pore radius Eq. (12)

and since the normalized permeability coefficient is related to the

water permeability coefficient of unstressed plain concrete through

the previously defined factors F and S, describing, respectively, the

influence of fiber reinforcement and stress Bhargava & Banthia

[27], the effective pore radius can be modified to:

effnormalized rSFr 25.025.0* (17)

where, r* normalized is the effective pore radius corresponding to

normalized permeability values and effr in this case is the effective

pore radius of plain concrete under zero stress condition.

Substituting Eq. (17) into Eq. (16), we get a modified equation

which relates normalized water permeability to diffusion

coefficient as,

14

DSCFKnormalized5.05.0 (18)

where C =

o

eff

D

gr

8

2

is a constant proportional to second power of the

effective pore radius of plain concrete under zero stress condition.

For plain concrete and zero stress condition F=S=1 and for this

case:

CxDKKunstressedplainwnormalized

(19)

Substituting the empirical values of the water permeability

coefficient Kw=1.07x10-10

m/s and the chloride ion diffusion

coefficient D = 7.89x10-13

m2/s, as obtained previously, the value

of constant C for the concrete in question can be calculated:

C = 135.62 m-1

(20)

The constant C computed above takes into consideration the

effective pore radius of plain concrete under zero stress condition

and properties of the chloride ion diffusion coefficient. The

calculated chloride ion diffusion coefficients are given in Table 3.

The Durability Factor, D, for a given concrete under a given stress

level can be defined as the ratio of its expected service life to that

of companion plain concrete under zero stress. Using Tuutti‘s

model [29], ingress of chlorides is estimated by a one-dimensional

diffusion process using the Fick‘s Second Law of diffusion. For

non-steady state condition, the chloride concentration C at a

location x and at a time t is given by Crank [33].

x

CD

xt

C

(21)

Here, the diffusion coefficient D may be a constant or a function of

other variables such as chloride concentration, location, time,

temperature, etc.

15

For a simple case with known geometry and boundary conditions

where the diffusion coefficient D can be assumed to be a constant,

solution to Eq. (21) is given by Newman [34]:

Dt

xerfCtxC s

21),( (22)

zt dtezerf

0

22)(

(23)

where,

erf is a standard error function,

x is effective concrete cover depth,

sC is the concentration of the chloride ions at the outside surface

of the concrete and is assumed to be constant with time. That is,

sCC for x = 0 and for any t

iC is the concentration at the depth of the reinforcement; assumed

to be zero at t =0.

tC is the threshold concentration required to initiate steel

reinforcement corrosion. The initiation period is accomplished

when ti CC and,

t = time

Eq. (22) can be solved by using a normal standard distribution

Bertolini et al. [35]:

1)2(2)( zNzerf (24)

dtezNz

t

2

2

2

2

1)2(

(25)

The initiation time can thus be calculated by assuming a constant

diffusion coefficient for concrete, a known surface chloride content

(dictated by the environment), the thickness of the concrete cover

and critical chloride ion content at which onset of corrosion is

expected.

16

Solving the above equation for tC = threshold concentration of

chloride ions = 0.50 % (based on the mass of cement), sC =chloride

ions concentration at the surface of concrete = 0.70 % (based on

the mass of cement), x = 25 mm, and diffusion coefficients, D,

from Table 1:

D2678.0

xtt

2

i (26)

Notice that a lower value of 0.50% threshold concentration of

chloride ions was chosen due to the presence of fly-ash in concrete

which is known to increase the rate of corrosion. The above

equation predicts that service life of any concrete is proportional to

x2, and holds an inverse relationship with the chloride ion diffusion

coefficient. Therefore doubling the concrete cover increases

service life of concrete by a factor of 4, whereas a 10-fold

reduction in diffusion coefficient will result in a 10-fold increase in

the predicted service life. Substituting the values of diffusion

coefficient from Table 3 into Eq. 26 for different concrete types

and stress conditions, the Durability Factors were computed and

are plotted in Fig. 16. Notice in Figure 16 that as per the model,

fiber reinforcement can be effective in enhancing the durability of

concrete under both stressed and unstressed conditions.

CONCLUDING REMARKS

A brief state-of-the-art report on fiber reinforced concrete is

presented. Our understanding of fiber-matrix interaction,

reinforcement mechanisms and performance characteristics is fairly

advanced. Fiber reinforced concrete is a promising material to be

used in the Middle-East for sustainable and long-lasting concrete

structures. Its performance has already been proven in other hot and

arid climates and in other chemically deleterious environments.

17

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Cementitious Composites under Impact Loading, Part 2: Flexural

Toughness, American Concrete Institute, Materials Journal, Vol. 98(1): pp.

17-24; 2001.

[22]. Kaadi, G.W., MS Thesis, The University of Illinois, Chicago, (1983).

[23]. Banthia, N., Impact and Blast Protection with Fiber Reinforced Concrete,

Conference Proceedings - BEFIB, Veronna, Italy, RILEM, 39, pp. 31-44;

2004.

[24]. Armelin, H. and Banthia, N., ACI Mat. J., 94(1): pp. 18-31; 1997.

[25]. Bentur, A., et al. 2005. Comprehensive Approach for the Design of

Concrete for Durability and Long Term Performance of Structures.

ConMat05 Mindess Symposium Proc., University of British Columbia (Ed.

Banthia, Bentur and Shah), 10 pp.

[26]. Banthia, N. and Bhargava, A., 2007. American Concrete Institute, Materials

Journal, 104(1), pp. 303-309.

[27]. Bhargava, A. and Banthia, N. 2008. RILEM, Materials and Structures, 41:

363-372.

[28]. Edvardsen, C., 1999. ACI Materials Journal, 96(4): 448-454.

19

[29]. Tuutti, K., 1982. Corrosion of steel in concrete. Swedish Cement and

Concrete Research Institute, Stockholm, Sweden. CBI Research Report no.

4.82.

[30]. Hedegaard, S.E. and Hansen, T.C., 1992. Materials and Structures, 25: 381-

387.

[31]. Hansen, T.C., Jensen, J., Johannesson, T. 1986. Cement and Concrete

Research, 16(5): 782-784.

[32]. Garboczi, J., Cement and Concrete Research, 20 (4) (1990) 590-601.TITLE

[33]. Crank J., ―Mathematics of diffusion‖, Oxford: Clarendon Press, 1956.

[34]. Newman, A.B. 1970. American Institute of Chemical Engineers, Vol. 27

pages.

[35]. Bertolini, L., Elsener, B., Pedeferri, P., and Polder, R., WILEY-VCH Verlag

GmbH and Co. kGaA, Weinheim (2004).

20

TABLES

Table 1. Properties of Fibers used as Reinforcement in Concrete

Fiber type Tensile

strength

(MPa)

Tensile

modulus

(GPa)

Tensile

strain (%)

(max-min)

Fiber

diameter

(m)

Alkali

stability,

(relative)

Asbestos 600-3600 69-150 0.3-0.1 0.02-30 excellent

Carbon 590-4800 28-520 2-1 7-18 excellent

Aramid 2700 62-130 4-3 11-12 good

Polypropylene 200-700 0.5-9.8 15-10 10-150 excellent

Polyamide 700-1000 3.9-6.0 15-10 10-50 -

Polyester 800-1300 up to 15 20-8 10-50 -

Rayon 450-1100 up to 11 15-7 10-50 fair

Polyvinyl

Alcohol 800-1500 29-40 10-6 14-600 good

Polyacrylonitrile 850-1000 17-18 9 19 good

Polyethylene 400 2-4 400-100 40 excellent

Polyethylene

pulp (oriented) ---------- ---------- ---------- 1-20 excellent

High Density

Polyethylene 2585 117 2.2 38 excellent

Carbon steel 3000 200 2-1 50-85 excellent

Stainless steel 3000 200 2-1 50-85 excellent

AR- Glass 1700 72 2 12-20 good

21

Table 2. Description of Test Methods

Standards ASTM C 1609-08 JSCE SF-4 ASTM C 1399-98

Test

Specimen

Reloading

Test

Description

A beam specimen is quasi-statically loaded at its third-points to

failure and the resulting load vs. net center point deflection is

plotted for further analysis.

A stable narrow crack is first created in the specimen by applying a

flexural load in series with a steel plate under controlled conditions.

The plate is then removed, and the specimen is reloaded in flexure to

obtain the post-crack load vs. net displacement curve.

Typical

Curve

Net Deflection, mm

Lo

ad

, N

First Crack

o

A

B C DE

F G Hδ 3δ 5.5δ 10.5δ

I J

δtb = L/150

Net Deflection, mm

Lo

ad

, N

Initial Loading Curve

Reloading Curve (Pre-cracked Beam)

P0.5 P0.75 P1.0 P1.25

Analysis

Px,y = Load at displacement

y for a x mm section

f150,0.75 (MPa): Residual

strength at P150,0.75

f150,3.0 (MPa): Residual

strength at P150,3.0

Toughness 150,3.0 (J): The

energy to a net deflection of

1⁄150 of the span (3.0 mm

for a 150 mm specimen)

Flexural Toughness (Tb) = Area OAEJ

Flexural Toughness Factor (FT)

FT = 2

tb

b

bd x

L x T

MOR = Modulus of Rupture

b = Breadth of the Beam

d = Depth of the Beam

Average Residual Strength

ARS = ((P0.5+P0.75+P1.0+P1.25)/4) x L/bd2

Initial Loading

100(%)2Re xMOR

FT

22

Table 3. Computed Values of Chloride Ion Diffusion Coefficient

Fiber

Volume

Fraction

Vf

Applied

Stress

Level

Normalized

water

permeability

coefficient

Knormalizedx10-

10 (m/s)

F

S

Chloride

ion

diffusion

coefficient

Dx10-13

(m2/s)

0.0%

0.0fu 1.66 1 1 12.24

0.3fu 103 1 0.62 9.64

0.5fu 2.30 1 1.38 14.43

0.1%

0.0fu 0.95 0.57 1 9.27

0.3fu 0.53 0.57 0.57 6.85

0.5fu 0.71 0.57 0.76 7.95

0.3%

0.0fu 0.60 0.36 1 7.37

0.3fu 0.32 0.36 0.53 5.40

0.5fu 0.45 0.36 0.75 6.38

0.5%

0.0fu 0.30 0.18 1 5.21

0.3fu 0.10 0.18 0.33 3.02

0.5fu 0.18 0.18 0.62 3.97

23

0.0

0.5

1.0

1.5

2.0

2.5

0.0% 0.1% 0.2% 0.3% 0.4%Volume Fraction (%)

Avera

ge C

rack W

idth

(m

m)

F1 F2

F3 F4

F5 F6

F7

FIGURES

Fig. 1. (Left) Plastic Shrinkage Crack Control Efficiency with

Increasing Fiber Volume Fraction from Top to Bottom and (Right)

Maximum Crack Width for Various Fibers (F1-F7).

Fig. 2. Fiber Reinforcement Before and After the Creation of

a Macro-Crack (Left) and Crack Bridging by Fibers (Right).

Micro crack

Formation

Macro crack

Formation

peak

A

B

A

A

B

A

24

Fig. 3. (Left) A CFRC Composite in Tension and (Right) Stress-Strain

Curves Showing Strain-Hardening at High Fiber Volume Fractions.

Fig. 4. (Left) A Fiber Pull-Out Test and (Right) Bond-Slip Pull-Out

Curves for Various Deformed Fibers. (Notice fiber fracture in an

excessively deformed fiber.)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 5 10 15Slip (mm)

Lo

ad

(kN

)

Undeformed

Deformed Fiber (Pull-Out)

Excessively Deformed Fiber (Fracture)

25

Fig. 5. Shear-Lag in a Bonded

Fiber with Inelastic Mechanisms

.

Fig. 6. R-Curve Representation of

Fracture in FRC.

Fig. 8. R-Curves Generated from CDCB Tests

Shown in Figure 7

Fig. 7. A CDCB Before and After Fracture.

26

Fig. 9. Impact Resistance of Steel FRC and Polypropylene FRC. (Note

the increase in brittleness in SFRC at high rates of loading.)

Fig. 10. Impact Response of SFRC Beams.

(Notice brittleness at high strain-rates.)

0

10

20

30

40

50

60

70

80

90

200 500 750 1000 Drop Height (mm)

To

ug

hn

es

s (

Nm

) Steel Fiber Polypropylene Fiber

0

50

100

150

200

250

300

350

0 0.2 0.4 0.6 0.8 1

Deflection (mm)

Lo

ad

(k

N)

Low Strain-Rate Impact

High Strain-Rate Impact

Quasi-Static

27

0

5

10

15

20

25

30

35

40

45

0 0.5 1 1.5 2 2.5 3

deflection (mm)

loa

d (

kN

)

Experimental

Analytical

Experimental Response Analytical Response

Peak Load

(kN) 38 34

JSCE

Toughness

Factor (MPa)

6.30 7.05

Fig. 11. Schematic View of Forces and Stresses Acting on the

Cracked Section of an SFRC Beam.

Fig. 12. Model Predictions under Impact Loading. (Note that

predicted response is monotonic, but predicts both peak load and

toughness.)

0

0

dis

pla

cem

ents

f i.. f

3 f 2

f 1

ƒ i = f(w i , i , l i )

c

uncr

acked

sect

ion

w i

1

2

3 i

stra

ins

28

Fig. 13. Effect of Crack Width on Permeability [1].

Fig. 14. The Effect of Crack Width on Corrosion Potential;

a Potential below –280mV Indicates Corrosion Initiation

and below –400mV Indicates Active Corrosion.

29

0

0.5

1

1.5

2

2.5

0 0.1 0.2 0.3 0.4 0.5

Stress Level ( f u )

Norm

alize

d P

erm

eability C

oeff

icie

nt x 1

0-1

0

(m/s

)

0.0% Fiber

0.1% Fiber

0.3% Fiber

0.5% Fiber

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.2 0.4 0.6

Stress Level (f u )

Du

rab

ilit

y F

act

or,

D 0.0% Fiber

0.1% Fiber

0.3% Fiber

0.5% Fiber

.

Fig. 15. Normalized Permeability Coefficients.

Fig. 16. Durability Factors. (Notice durability enhancements with

fibre reinforcement.)