FIBER REINFORCED CONCRETE FOR LIFE TIME ENGINEERING OF CIVIL INFRASTRUCTURE

N. Banthia

University of British Columbia, Canada

ABSTRACT: 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. With the insurmountable durability concerns for our current transportation infrastructure and especially bridge decks, the use of high performance fiber reinforced concrete for both new infrastructure and for rehabilitation of older infrastructure appears highly promising. Fibers control shrinkage cracking, abate micro-cracks from widening and provide concrete with high ductility, toughness, impact resistance and fatigue endurance. Fibers also reduce permeability of stressed concrete and can be made to undergo multiple cracking such that the resulting material is significantly more damage tolerant while maintaining its stress carrying capability. This paper presents a brief state-of-the-art report on mechanical properties and durability of fiber reinforced concrete and identifies the numerous advantages that may be potentially derived from use of such high performance materials. KEYWORDS: FRC, fiber reinforced concrete, toughness, fatigue, shrinkage, durability, permeability, bond, repair 1. INTRODUCTION Compared to other building materials such as metals and polymers, concrete is significantly more brittle and exhibits a poor tensile strength. 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 tips of the macro-cracks and unstable and catastrophic failure is precipitated. Under fatigue loads, concrete cracks easily, and cracks create easy access routes for deleterious agents leading to early saturation, freeze-thaw damage, scaling, discoloration and steel corrosion. The micro and macro-fracturing processes described above can be favourably 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. The resulting material termed fiber reinforced concrete (FRC) is rapidly becoming a well-accepted mainstream construction material. There are currently 200,000 metric tons of fibers used for concrete reinforcement. Table 1 shows the existing commercial fibers and their properties. This paper discusses the use of fiber reinforced concrete in transportation infrastructure especially bridge decks. 2. FIBER REINFORCEMENT MECHANISMS 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 (Figure 1). If the fiber volume fraction is sufficiently high, this may result in an increase in the tensile strength of the matrix beyond the Bend Over Point, BOP. Indeed, for some high volume fraction fiber composite [1], a notable increase in the tensile/flexural strength over and above the plain matrix has been reported (Figure 2). 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 mechanisms in the majority of commercial fiber reinforced concrete composites.

Figure 1. Fiber Reinforcement Before and After the Creation of a Macro-Crack (Left) and Crack Bridging by Fibers (Right)

Table 1. Properties of Fibers used as Reinforcement in Concrete

Fiber type Tensile strength (MPa)

Tensile modulus

(GPa)

Tensile strain ( %) (max-min)

Fiber diameter

(μm)

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 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

3. CRITICAL FIBER VOLUME, STRAIN HARDENING AND MULTIPLE CRACKING It emerges therefore 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 normal performance 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 either a reduction in the plastic shrinkage crack control or to enhancement of energy absorption (‘toughness’) in the post-cracking regime only. For high performance fiber reinforced composites, on the other hand, with high fiber dosages, benefits of fiber reinforcement are noted in an increased tensile strength, strain-ha.rdening response before localization and enhanced ‘toughness’ beyond crack localization. Fiber volume fraction at which fibers can be expected to produce an increase in the tensile/flexural strength is given by [2]:

Micro crack Formation

Macro crack Formation

σ

ε

ε peak

BOP

Pre-BOP Post-BOP

)(1

1)(

21321 ααλλλστ

−+=≥

f

f

mu

fucriticalff

dlVV (1)

where, τfu is the average interfacial bond strength at the interface, σmu is the tensile strength of the matrix, lf is the fiber length and df is the fiber diameter. λ1, λ2, λ3 are efficiency factors related to length, orientation and grouping, respectively, and α1 and α2 are constants pertaining to uncracked state of the composite. For a given FRC, Equation 1 guarantees that if the critical volume fraction is exceeded, composite will depict strain hardening and show multiple cracking. Some such curves for carbon fiber reinforced concrete in tension are given in Figure 2. Note the presence of strain hardening in the composite beyond 2% by volume.

Figure 2. (Left) A CFRC Composite in Tension and (Right) Stress-Strain Curves

Showing Strain-Hardening and Multiple Cracking at High Fiber Volume Fractions In FRCs with volume fractions higher than the critical, after the bend-over point, BOP, (Figure 2), multiple cracking is expected to occur and the composite is expected to crack in segments of lengths between x and 2x (where x is the transfer length) given by:

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

42 f

fu

mu

f

m dVVx

τσ (2)

4. FRACTURE MECHANICS BASED REPRESENTATION Equation (1) is a rather simplistic approach to FRC. It assumes that the composite goes from an uncracked state to a cracked state (albeit showing multiple cracking at Vf > Vfcritical). In reality, concrete is a micro-fracturing, strain-softening material, and 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 complex and advanced models are needed to simulate these processes. Attempts have been made to model fracture in FRC using the cohesive crack model [3] as well as the J-integral [4]. However, strictly speaking, these are only crack initiation criteria 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 [5]. An R-curve (Figure 3) 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. Some R-Curves are shown in Figure 4 [6].

Figure 3. R-Curve Representation of

Fracture in FRC Figure 4. R-Curves Generated from

A Crack Growth Test 5. FIBER-MATRIX INTERFACIAL BOND As in any fiber reinforced composite, fiber-matrix bond in FRC is of critical importance. However, unlike fiber reinforced polymers (FRPs) with continuous fibers used by the aerospace and automobile industries, short fibers in FRC mean that the bond in most cases is not fully developed. For a fiber embedded in a cementitious matrix and subjected to a pull-out load (Figure 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. Tensile stress in the fiber [σf (x)] and shear stress at the interface [τ (x)] can be given by:

(3)

where R = Radius of matrix around the fiber taking part in transfer, r = radius of the fiber, lf = length of the fiber, Ef = modulus of elasticity of the fiber, Gm = Shear modulus of the matrix at the interface, R/r depends upon the fiber packing and fiber volume fraction. For 2-D packing: ln R/r = (1/2) ln (p/Vf) and for 3-D packing: ln R/r = (1/2) ln [2p/(3Vf)½].

( ) mff

f

f El

xl

x εβ

βσ ⋅

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−

=

2cosh

)2

cosh1

1

1

2/1

21

ln

2

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛

=

rRrE

G

f

mβ

( )

2cosh

21sinh

ln2 1

1

)2/1(

ff

mmff l

x

rRE

GEx

β

βετ

⎟⎠⎞

⎜⎝⎛ −

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛

=

One can show that the ratio of the maximum interfacial bond stress [τ (max)] to the tensile stress [σf (max)] is given by:

4coth

)/ln(2(max)(max) 1

21

f

ff

lrRE

G βστ

⎥⎥⎦

⎤

⎢⎢⎣

⎡= (4)

For 2% fiber volume, with a τmax = 15 MPa one gets a σmax = 200 MPa which is much lower than the yield strength of steel. Consequently, a steel fiber for normal lengths of 25-60 mm cannot ever develop stresses close to its capacity and hence most fibers in practice are deformed. However, even here there is a limit. If deformed excessively, fibers may develop stresses that exceed their strength and fracture in the process (Figure 6). 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. In FRC, inelastic bond failure mechanisms such as interfacial crack growth, crack tortuousity and fiber slip are of greater relevance.

Figure 5. Shear-Lag in a Bonded Fiber

with Inelastic Mechanisms Figure 6. Bond-Slip Pull-Out Curves for Various Deformed

Fibers. Notice Fiber Fracture in an Excessively Deformed Fiber

6. SOFTENING AND TOUGHNESS In the softening regime, where the load starts to drop, the response of the composite is completely dependent upon the bond-slip behaviour of the fibers under an applied pull-out load. The response in the softening regime can therefore be assembled by first expressing the bond-slip behavior of a given fiber and then integrating the contribution of all fibers across a crack. In the case of pure tension, the stress vs crack separation, σtension(w), curve can be expressed as [7]:

αααπ

σπ

α

αdzdzppwf

dV

w fL

zf

ftension )()(),(

4/)(

0

cos)2/(

02 ∫ ∫= == (5)

where, f(α, w) is the bond-slip response of a single fiber at a crack opening (w) and inclination angle α, p(α) and p(z) are probability density functions of the orientation of fiber w.r.t. the tensile loading direction and centroidal distance of the fiber from the crack plane, respectively.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 5 10 15Slip (mm)

Load

(kN

)

Undeformed

Deformed Fiber (Pull-Out)

Excessively Deformed Fiber (Fracture)a) Intact interface.

b) Partially debonded interface.

Figure 7. Modeling FRC in Flexure In the case of flexure, which is more pertinent to FRC, Armelin and Banthia [8] proposed a stochastic model (Figure 7). The compressive strain, εo, at the top-most fiber of the specimen leads to an axial shortening, Δo, as shown. This in turn leads to stress, σc, in the 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:

lM

P e2= (6)

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 Figure 7, that

( )∫ ∑ =+'

0 10.

c N

ic fdybσ (equilibrating forces) (7)

( ) ( )∑∫ +=N

ii

c

ce yfydybM10

.'

σ (equilibrating moments) (8)

The model expresses the pull-out force in each fiber (fi) 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.

( )( )[ ] ⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

+

−+=

CCi

ipiii

Bw

AAwEwf 1

1

1,α (9)

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 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’ to be substituted in Equations 7 and 8 may be computed as:

( ) ( ) ( ) ( ) ( ) ( )⎭⎬⎫

⎩⎨⎧

+⎥⎦⎤

⎢⎣⎡ ++++= wf

wfwfwfwf

wff geometryi 4

1222

1 905.67455.22

0 (10)

CMOD

0

ε 0

disp

lace

men

ts

f i.. f 3

f 2 f 1

ƒi = f(w i , α i , l i )

σc

uncr

acke

d

sect

ion

w i

12

3i

stra

ins

C’

οΔ

Some predictions of the above model are compared with experimental findings in Figure 8. Note an excellent match.

Figure 8. Model Predictions for FRC Flexural Toughness

7. DYNAMIC PROPERTIES Due to the excellent ability of fibers to control crack growth and provide crack-tip toughening, the fatigue performance of concrete is significantly enhanced due to fiber reinforcement (Figure 9) [9]. Drop Weight Impact Tests [10] are generally performed to measure the resistance of fiber reinforced concrete to impact loads. For fiber reinforced concrete, while an improvement in impact properties is widely reported, on a worrisome note, steel fibers are reported [11] to fracture across cracks at high rates of loading and thus produce a brittle response at very high strain-rates. As seen in Figures 10 and 11, SFRC may show increased brittleness under very high strain rates. Polypropylene fibers, on the other hand, do not show onset of brittleness at high rates of loading. The exact reasons of the observed brittleness of some FRC materials under impact have been investigated via fundamental testing of bond-slip mechanisms, fracture studies and modeling [12].

Figure 9 Fatigue Response of FRC

Figure 10. Impact Resistance of Steel FRC and Polypropylene FRC. Note the

increased brittleness in SFRC at high rates of loading.

Figure 11. Impact Response of SFRC Beams. Notice Brittleness at

High Strain-Rates 8. SHRINKAGE Soon after placement, evaporation of the mix water and the autogenous process of concrete hydration cause shrinkage strains in concrete. With their 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 (Figure 12a; Ref. 13). Further, when the concrete is restrained, as will be the case in a bridge deck, fibers bridge cracks and reduce crack widths and crack areas (Figure12 b, c and d). Indeed, a number of attempts have been made in the past to provide shrinkage and thermal reinfor cement in bridge decks using fiber reinforcement as in the ‘steel free’ deck systems [14, 15, 16].

0 10 20 30 40 50 60 70 80 90

200 500 750 1000 Drop Height (mm)

Toug

hnes

s (N

m) Steel Fiber

Polypropylene Fiber

0

50

100

150

200

250

300

350

0 0.2 0.4 0.6 0.8 1Deflection (mm)

Load

(kN

)

Low Strain-Rate Impact

High Strain-Rate Impact

Quasi-Static

0.0

0.5

1.0

1.5

2.0

2.5

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

Ave

rage

Cra

ck W

idth

(mm

)

F1 F2F3 F4F5 F6F7

0

50

100

150

200

250

300

350

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

Ave

rage

Cra

ck A

rea

(mm

2 )

F1 F2F3 F4F5 F6F7

Figure 12 a. Free Shrinkage Strains in FRC and Plain Concrete.

Figure 12 b. Control of Shrinkage Cracking in Restrained Overlay (Top: Plain; Middle: 0.1% Fiber; Bottom:

0.2% Fiber

Figure 12 c. Plastic Shrinkage Crack Control Efficiency of Various Fibers (F1-F7): Average

Crack Area

Figure 12 d. Plastic Shrinkage Crack Control Efficiency of Various Fibers

(F1-F7): Average Crack Width 9. PERMEABILITY, CRACKING AND SERVICE LIFE PREDICTIONS Permeability and Cracking: The long term performance of bridge decks is becoming an issue of greater significance in modern bridge engineering. In this context, corrosion of the reinforcing steel is the biggest concern. 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 bridge decks. 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 mode. Any measures adopted to reduce permeability of concrete will therefore help in preserving the durability of a concrete deck. 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 (Figure 13) but also enhance the possibility of rebar corrosion (Figure 14) [17].

Figure 13: Effect of Crack Width on Permeability [17]

Figure 14. Effect of Crack Width on Corrosion Potential. A potential below –280mV indicates

corrosion initiation, and below –400mV indicates active corrosion [17]

Bentur et al [17] also considered the potential of fibers and other technologies in controlling crack widths in a typical bridge deck and their results are given in Table 2. It was found that the use of shrinkage reducing admixture and fibers could be as effective as doubling of the steel reinforcement.

Table 2: Effect of the Type of Reinforcement on Cracking in a 30m Long Bridge Deck (w/cm = 0.38; FEMASSE Software)

Crack width, mm Reinforcement

Average Minimum Maximum Number of

cracks Rebar 0.40 0.25 0.60 15 Doubling of rebar 0.21 0.11 0.25 22 Rebar + SRA1 0.22 0.18 0.28 8 Rebar + fiber2 0.29 0.20 0.36 23

1 Shrinkage Reducing Admixture 2 Equivalent flexural strength of 1.8 MPa (JCI-SF4)

0

0.5

1

1.5

2

2.5

0 0.1 0.2 0.3 0.4 0.5

Stress Level ( f u )

Nor

mal

ized

Per

mea

bilit

y C

oeff

icien

t x 1

0-10

(m/s)

0.0% Fiber0.1% Fiber0.3% Fiber0.5% Fiber

The influence of an externally applied stress on the permeability of concrete remains poorly understood. Banthia and co-workers [18, 19] 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:

hAQLK w Δ

= (11)

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 Δh = Pressure head (m) Their data are plotted in Figure 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.

Figure 15. Normalized Permeability Coefficients 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 [20], 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. Service Life Prediction: Bhargava and Banthia [19] extended the permeability data described above towards service life prediction. Most service life prediction models for concrete involve the use of diffusion coefficients [21]. 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. [22] and for diffusion coefficient were proposed by Hansen et al. [23] as follows:

⎥⎦

⎤⎢⎣

⎡+⎟

⎠⎞

⎜⎝⎛ +

−= 0.431.03.4expw

fcKw (12)

⎟⎠⎞

⎜⎝⎛ +

+−

=0.73.0

107.1 wfc

xD (13) where, Kw= water permeability coefficient (m/s) D = Chloride ion diffusion coefficient, in cm2/s c = cement content of concrete, in kg/m3 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 Equations 12 and 13, 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.

Ar

K eff

8

4π= (14)

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,

Ar

DaDD effoeffo

2π== (15)

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 Equations 14 and 15 is the same, a general relationship between permeability K (m2) and diffusion coefficients D (m2/s) emerges,

DD

rK

o

eff

8

2

= (16)

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,

g

KK w

ρη

= (17)

Using Equations 16 and 17, the water permeability coefficient wK (m/s) and the diffusion coefficient D (m2/s) can be related as,

DD

grK

o

effw η

ρ

8

2

= (18)

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

D is the diffusion coefficient (m2/s), reff is the effective pore radius, η is the viscosity of water (Ns/m2), ρ is the density of water (kg/m3) and, g is the gravity (m/s2)

This equation corresponds to Katz-Thompson Equation [24], and is based on the assumption that the effective radius affecting the permeability and the diffusion coefficient is the same. Equation 18 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 (Equation 14) 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 [see Ref. 19 for details], the effective pore radius can be modified to:

effnormalized rSFr 25.025.0* = (19)

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 Eqn 19 into Eqn. 18, we get a modified equation which relates normalized water permeability to diffusion coefficient as,

DSCFKnormalized5.05.0= (20)

where C =η

ρ

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:

CxDKK unstressedplainwnormalized == − (21)

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 (22) 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.

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)

0.0fu 1.66 1 1 12.24 0.3fu 103 1 0.62 9.64

0.0%

0.5fu 2.30 1 1.38 14.43 0.0fu 0.95 0.57 1 9.27 0.3fu 0.53 0.57 0.57 6.85

0.1%

0.5fu 0.71 0.57 0.76 7.95 0.0fu 0.60 0.36 1 7.37 0.3fu 0.32 0.36 0.53 5.40

0.3%

0.5fu 0.45 0.36 0.75 6.38 0.0fu 0.30 0.18 1 5.21 0.3fu 0.10 0.18 0.33 3.02

0.5%

0.5fu 0.18 0.18 0.62 3.97 In this study, the Durability Factor, D, for a given concrete under a given stress level was defined as the ratio of its expected service life to that of companion plain concrete under zero stress. Using Tuutti’s model [21], 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 [25].

⎟⎠⎞

⎜⎝⎛=

xCD

xtC

δδ

δδ

δδ

(23)

Here, the diffusion coefficient D may be a constant or a function of other variables such as chloride concentration, location, time, temperature, etc. 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. 23 is given by [26]:

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−=

DtxerfCtxC s 2

1),( (24)

∫−=

z t dtezerf0

22)(π

(25)

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.24 can be solved by using a normal standard distribution [27]:

1)2(2)( −= zNzerf (26)

dtezNz

t

∫∞−

=2

2

2

21)2(π

(27)

00.5

11.5

22.5

33.5

44.5

0 0.2 0.4 0.6Stress Level (f u )

Dur

abili

ty F

acto

r, D 0.0% Fiber

0.1% Fiber0.3% Fiber0.5% Fiber

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. 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 3:

Dxtt i 2678.0

2=≈ (28)

Figure 16. Durability Factors: Notice Durability Enhancements with Fiber Reinforcement

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. 28 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. 10. FRC IN REPAIR: BOND WITH OLD CONCRETE A thin bonded concrete overlay provides an increase in the structural capacity, and rehabilitation of old bridge decks with FRC has become a common practice around the world. Any repair performed on a structure must meet four major requirements. First, it should be able to arrest further deterioration and particularly the corrosion of the reinforcing steel if present. This requires that the material used for repair be adequately impermeable to aggressive liquids and gases. The second requirement is that the repair material should be able to bond properly with the old concrete and restore structural integrity. Third, the repair should be durable and be able to withstand the severe climatic conditions imposed upon it. Finally, the repair material should have chemical, electrochemical, permeability and dimensional compatibility with the substrate. Based on the above, it is clear that fiber reinforced concrete has all of the attributes needed for a durable repair of bridge decks [28]. Indeed closed-loop repair bond tests with plain and fiber reinforced concrete [29] have indicated that there is both an increase in the bond strength and bond toughness as quantified by interfacial Gf values (Table 4) due to fiber reinforcement. Some typical bond strength curves are given in Figure 17.

Figure 17. Closed-Loop Repair Bond Strength Test (Left) and Results (Right). Notice the Beneficial Effect of Fibers

Table 4. Bond Strength of FRC with Old Concrete Surface Condition

Polymer Fiber Type of Failure

Interfacial Bond Strength

Interfacial Gf

Type Volume Fraction

(MPa) (N-mm/mm2) x102

0 % A * 0.74 0.019 Steel 1 % C ** 1.33 0.051 0 % 2 % A & C 0.97 0.020 Smooth carbon 1 % A & C 0.83 0.041 2 % C 1.13 0.063 0 % A 0.98 0.037 10 % Steel 1 % C 1.24 0.054 2 % C 1.49 0.078 0 % C 1.31 0.047 Steel 1 % C 1.75 0.085 Rough 0 % 2 % C 2.02 0.126 carbon 1 % A 1.03 0.055 2 % C 1.21 0.078 * Adhesive failure ** Cohesive failure 11. CONCLUDING REMARKS With the current durability concerns for our transportation infrastructure (and especially the bridge decks), the use of high performance fiber reinforced concrete for both new construction and repair appears highly promising. Fibers control shrinkage cracking, abate micro-cracks from coalescing and enhance ductility, toughness, impact resistance and fatigue endurance. With their high resistance to crack nucleation and growth, fibers reduce the permeability of concrete and prevent the ingress of deleterious agents thereby delaying both material degradation and steel corrosion.

12. REFERENCES

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Concepts (Ed. C. Vipulanandan), SP-201, pp. 55-74; 2000. [7] Li, V., , Kluwer Academic Publishers, The Netherlands, 1991, pp. 447-466. [8] Armelin, H. and Banthia, N., ACI Mat. J., 94(1): pp. 18-31; 1997. [9] Ramakrishnan, V., Proceedings of the Sixth International Purdue Conference on Concrete Pavement:

Design and Materials for High Performance, Indianapolis, Indiana, Nov., 18-21, 1997, pp. 119-130. [10] Banthia, N., Mindess, S., Bentur, A. and Pigeon, M., Expt. Mech. 29 (2): pp. 63-69; 1989. [11] Bindiganavile, V and Banthia, N., American Concrete Institute, Materials Journal, Vol. 98(1): pp. 17-24;

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[14] Newhook, J. P., and Mufti, A.A., Concrete International, V. 18, No. 6, 1996. [15] Mufti, A., Banthia, N. and Bakht, B., Banthia, N., Sakai, K. and Gjφrv, O.E, Proc., 3rd International

Conference on Concrete Under Severe Conditions of Environment and Loading,. (Eds.) Vancouver, June 2001, The University of BC, 2001, pp. 1032-1041.

[16] Banthia, N., Yan, C., Mufti, A., and Bakht, B., (Eds.: Peled, Shah and Banthia) ACI, Special Technical Publication, SP-190, American Concrete Institute, Detroit, USA, pp. 21-39.

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Weinheim (2004). [28] Carter, P., et al., Concrete International, July 2002 , pp. 51-58. [29] Banthia, N. and Yan, C., ACI Special Publication on High Performance Materials for Repairs (edited by

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